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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 46, pp. 1--42.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/46\hfil A hyperbolic-parabolic system]
{A hyperbolic-parabolic system arising in pulse combustion:
existence of solutions for the linearized problem}
\author[O. Terlyga, H. Bellout, F. Bloom \hfil EJDE-2013/46\hfilneg]
{Olga Terlyga, Hamid Bellout, Frederick Bloom}
\address{Olga Terlyga \newline
Fermi National Laboratory,
Batavia, IL 60510, USA}
\email{terlyga@fnal.gov}
\address{Hamid Bellout \newline
Department of Mathematical Sciences,
Northern Illinois University,
DeKalb, IL 60115, USA}
\email{sabachir@hotmail.com}
\address{Frederick Bloom \newline
Department of Mathematical Sciences,
Northern Illinois University,
DeKalb, IL 60115, USA}
\email{bloom@math.niu.edu, Phone 815-753-6765}
\thanks{Submitted July 10, 2012. Published February 8, 2013.}
\thanks{This article is based on the Ph.D. dissertation
of the first author at Northern Illinois University.}
\subjclass[2000]{35M33, 35B65, 80A25}
\keywords{Pulse combustion; linear hyperbolic-parabolic system;
\hfill\break\indent Galerkin approximation; global solution}
\begin{abstract}
A mixed hyperbolic-parabolic system is derived for a lumped parameter
continuum model of pulse combustion. For a regularized version of the
initial-boundary value problem for an associated linear system, with
time-dependent boundary conditions, Galerkin approximations are used to
establish the existence of a suitable class of unique solutions. Standard
parabolic theory is then employed to established higher regularity for the
solutions of the regularized problem. Finally, a
priori estimates are derived which allow for letting the artificial
viscosity, in the regularized system, approach zero so as to obtain the
existence of a unique solution for the original mixed hyperbolic-parabolic
problem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
Pulse combustion is a process in which pressure, velocity, and temperature,
vary periodically with time; it was first observed by Rayleigh \cite{r1}.
A basic pulse combustor consists of a set of intake valves for air and
fuel, a combustion chamber, and a tailpipe from which the combustion
products are expelled as a consequence of the oscillating pressure field
in the chamber. As air and gas enter the chamber, combustion raises the
temperature and pressure; when the pressure rises above atmospheric
pressure, the valves begin to close, the air-fuel input is
reduced or stopped and the combustion products begin
flowing through the tailpipe, leading to a decrease in chamber pressure.
Once the chamber pressure falls below atmospheric pressure the valves
open to admit another fuel-air charge. The gaseous products in the
tailpipe execute oscillations which are superimposed on a mean flow and
which produce a periodic jet that issues from the open end of the
tailpipe; it is this periodic jet which drives the resulting impingement
heat transfer process (see figures \ref{fig1}, \ref{fig2} and \ref{fig3}).
A variety of models have been constructed to
simulate the behavior of pulse combustors, e.g.,
\cite{a2,b1,b4,c2,d1,d2,d5,d7,f1,g1,h1,k7,k8,k9,m4,p1,w1,z1}
and
some work has been done to examine factors influencing combustion
in the combustion chamber and the
tailpipe flow field in \cite{d3,d4,k10,s2}. For
analysis of the associated problem of acoustic oscillations in resonance
tubes one may consult \cite{d6,k3,k11,k12,m1,s4,y1}.
A description of the complex processes occurring in a typical pulse
combustor may be found in \cite{b1,f1}, e.g., when the chamber pressure
is equal to atmospheric pressure turbulence is present within a layer
separating the fresh air-fuel mixture and the residual gas from the
preceding cycle; this layer contains ignition nuclei and is broken up
into vortices which are carried into the fresh mixture thus igniting it
and producing a flame which transits the length of the combustion
chamber. Analytical models of pulse combustor operation which take into
account the full range of physical processes present are not feasible.
Most pulse combustion models attack the problem by writing down a
set of conservation laws for the ongoing combustion process and
presenting results, in graphical form, based on numerical analysis of
the governing system; such an approach is not illuminating if the
goal is the production of a combined model which would allow for making
qualitative (as well as quantitative) predictions of the effect of
varying pulse combustor physical and geometrical design characteristics.
With the exception of the work in \cite{f1},
none of the literature has attempted to deal with the influence of value
design and operation on pulse combustor performance including the
important issue of the qualitative behavior of the jet which issues from
the combustor tailpipe; a notable exception is \cite{a2} where, however, the
model (in its original form) assumes an instantaneous opening and
closing of the valves that is unrealistic and has the effect of inducing
a discontinuity in the mathematical model. Shortcomings in the model
presented in \cite{a2}
have been addressed in \cite{b6}. Recently
results were obtained by applying the technique of averaging, to the dynamical
system
generated by the lumped parameter pulse combustor model discussed in
\cite{b6} and were presented in \cite{b7}.
The lumped parameter model of pulse combustion, which
is found in \cite{a2}, while not dealing with all the
chemical kinetics processes involved, incorporates a realistic valve
dynamics submodel and is capable of producing closed-form approximations
for pressure and temperature variations in the chamber and velocity
oscillations in the tailpipe. The work in \cite{a2} begins with a statement
of energy balance and assumes that there are two uniform
regions in the combustion chamber, a `cool' zone consisting of the
reactants and a hot
zone containing the combustion products; these regions are separated by
a moving flame front. The model assumes a
spatially uniform pressure $p(t)$, ignores friction in the tailpipe,
does not account for heat loss from the chamber, and also assumes that
the combustion products in the tailpipe are incompressible.
Balance of
energy in \cite{a2} yields the equation
\begin{equation}\label{eq1.1}
\Big( \frac{c_v V_B}{R} \Big) \frac{dp}{dt} = h_R \dot{m}_R
+ \frac{\Delta H}{1 + r} \dot{m}_B - h_0
\bar{\rho} Av(t)
\end{equation}
where $c_v / R$ is the (approximate) constant ratio of specific heat (at
constant volume) to the gas constant for the air and fuel, $V_B$ is the
chamber volume, $A$ is the cross-sectional area of the cylindrical
tailpipe, $r$ is the air-fuel (mass) ratio, $\Delta H$ is the heat of
combustion per unit mass of fuel, $h_R$ is the enthalpy, per unit mass,
of the reactant mixture, $h_0$ is the enthalpy, per unit mass of the
combustion products, $v(t)$ is the velocity of the combustion products
in the tailpipe, $\bar{\rho}$ is the average density of the combustion
products in the tailpipe, and $\dot{m}_R, \dot{m}_B$ are, respectively,
the mass flow rate of the reactants and the mass burning rate of the
reactant mixture in the chamber. Coupled to \eqref{eq1.1}, as a consequence of
the continuity equation, is the balance of momentum equation
\begin{equation}\label{eq1.2}
\bar{\rho} L \frac{dv}{dt} = p(t) - p_a \equiv \tilde{p}(t)
\end{equation}
In \eqref{eq1.2}, $L$ is the length of the tailpipe
while $p_a$ is
atmospheric pressure at the open end of the tailpipe. To \eqref{eq1.1} and
\eqref{eq1.2} we
must append constitutive equations relating the reactant mass flow rate
$\dot{m}_R$ and the mass burning rate $\dot{m}_B$ to the chamber
pressure $p$. In \cite{a2} it was assumed that the valves are, at any time
$t$, either fully-open, or fully-closed, depending on whether or not there
exists a pressure induced driving force for flow into the chamber; these
inflows of air and fuel (gas), with respective mass flow rates
$\dot{m}_a$ and $\dot{m}_g$, were described by the orifice flow
equations
\begin{subequations}\label{eq1.3}
\begin{gather}
\dot{m}_a = \begin{cases}
\sqrt{2 \rho_a } C_{D{a}} A_a \sqrt{p_a - p}, & p < p_a \\
0, &p \geq p_a ,\end{cases} \\
\dot{m}_g = \begin{cases}
\sqrt{2 \rho_{g}} C_{Dg} A_g \sqrt{p_g - p}, & p < p_g \\
0, &p \geq p_g, \end{cases}
\end{gather}
\end{subequations}
where $C_{D_{a}}, C_{Dg}$ are the discharge coefficients of the air and
gas valves, $\rho_a$ and $\rho_g$ are the air and gas densities, and
$A_a$ and $A_g$ are the effective flow areas of the air and gas valves.
If $p_g \simeq p_a$ then
\begin{equation}\label{eq1.4}
\dot{m}_R = (1 + r) \dot{m}_g = \begin{cases}
(1 + r) \Gamma_g \sqrt{p_a - p}, & p < p_a \\
0, & p \geq p_a, \end{cases}
\end{equation}
where $r = \Gamma_a / \Gamma_g$ is the constant air-fuel ratio with
$\Gamma_g = \sqrt{2 \rho_g} C_{D_{g}} A_g$ and
$\Gamma_a = \sqrt{2 \rho_a} C_{D_{a}} A_a$.
The relation \eqref{eq1.4} presents two difficulties: (i) it assumes an
instantaneous opening (closing) of the valves at any time when the
combustion pressure $p(t)$ falls below (rises above) $p_a$ and
(ii) it yields an $\dot{m}_R(p)$ which is not differentiable at any $t$ where
$p(t) = p_a$; to deal with these problems the authors in \cite{a2}
replaced \eqref{eq1.4} by $\dot{m}_R^{\epsilon}
= \Gamma_g(1 + r) H_{\epsilon}(p - p_a) \sqrt{p - p_a}$ where $H_{\epsilon}$,
for $\epsilon > 0$, represents a smoothing
of the usual Heaviside function. Using this approach, one can then either
study the resulting model for finite $\epsilon > 0$, which leads to valve
hysteresis, or, by imposing stability criteria associated with
stable burner operation, extract explicit approximate expressions for
the frequency $\omega$ and period $T_p$, of combustor pressure
oscillations, as $\epsilon \to 0^+$;
these stability criteria are equivalent to the statements
\begin{itemize}
\item[(i)]
there should be no net reactant accumulation or depletion over one
cycle; i.e.,
\begin{subequations}
\begin{equation}
\label{eq1.5a}
\lim_{\epsilon \to 0^+} \int_0^{T_{p}} \dot{m}_R^{\epsilon} dt
= \int_0^{T_{p}} \dot{m}_B dt\,,
\end{equation}
\item[(ii)]
there should be no net pressure buildup or decay over successive cycles; i.e.,
\begin{equation}
\label{eq1.5b}
\lim_{\epsilon \to 0^+} \int_0^{T_{p}} (p_{\epsilon}(t) - p_a)
dt = 0\,.
\end{equation}
\end{subequations}
\end{itemize}
In \cite{a2} the actual flame structure in the chamber was idealized to
consist of an equivalent plane flame sheet filling the combustion
chamber cross-sectional area $A_B$; the plane flame propagates with a
`burning velocity' $U_f$, which is pressure-independent, relative to the
unburned reactant mixture in the chamber. Under these assumptions
$\dot{m}_B = \rho_R A_B U_f$ and, if the reactants (air and gas) are
taken to be at the same constant temperature $\theta_a$, the perfect gas
law yields $\dot{m}_B = (A_B U_f / R \theta_a)p$.
In \cite{b6} it was shown that the model described above allows for the
computation of analytical expressions for $p(t), v(t), U_f, T_p$ and the
velocity $v_0$ at the inception of the first full stable chamber cycle,
which display an explicit dependence on all relevant combustor physical
and geometrical parameters; for a range of air-fuel ratios $r$
these expressions yield a tailpipe velocity which exhibits flow
reversal. In this model $U_f$ is not the actual flame velocity, which
depends on the specific diffusion, heat transfer, and chemical kinetics
mechanisms at work during the burning process but is, rather, a system
parameter whose value is compatible with the achievement of stable
system oscillation.
Upon eliminating between \eqref{eq1.1} and \eqref{eq1.2} one obtains
for $\tilde{p}(t)$ the nonlinear second order equation
\begin{equation}
\label{eq1.6}
\frac{d^2 \tilde{p}}{d t^2} - \{ \lambda^u + \lambda^d (\tilde{p})
\} \frac{d \tilde{p}}{dt} + \omega_0^2 \tilde{p} = 0\,,
\end{equation}
where $\omega_0^2 = R h_{\theta} A / c_v V_B L$ and
\begin{equation}\label{eq1.7}
\lambda^u = \frac{\Delta H}{1 + r} \Big( \frac{A_B U_f}{c_v \theta_a
V_B} \Big) , \quad \lambda^d ( \tilde{p} ) = \Big( \frac{Rh_R}{c_v V_B}
\Big) \frac{d}{d \tilde{p}} \dot{m}_R ( \tilde{p} )\,.
\end{equation}
In \cite{b6} an approximation to the solution of the
initial-value problem for \eqref{eq1.6} was constructed which is periodic with period
\begin{equation}
\label{eq1.8}
T_p = \frac{\pi}{\omega_{\ast}} + \frac{\pi}{\sqrt{2 \omega_0}}
\Big( 1 + \exp [ \frac{- \lambda^u \pi}{2 \omega_{\ast}} ] \Big)^{1/2}
\end{equation}
where $\omega_{\ast} = \omega_0 \sqrt{1 - \frac{1}{4} \epsilon^2}$ with
$ \epsilon = \lambda^u / \omega_0$. For $\epsilon < < 1$
$T_p \approx 2 \pi / \omega_{\ast}$.
Further results have been obtained
recently by one of the authors and his colleagues, in \cite{b5,b7,b8,b9,b10,b11,b12}, by applying
perturbation theory and dynamical systems analyses to study the behavior exhibited by spatially
independent pulse combustor models of the type presented in this section;
these results relate, e.g., to the effect of tailpipe friction on pressure
and velocity oscillations, the influence of convective and radiative heat
transfer, and the optimization of reactant flow rates and mass burning rates
in lumped parameter pulse combustor models. For the balance of this paper,
as well as in the follow-up paper \cite{b3}, the focus will be on the pulse
combustor models incorporating spatial dependence.
In the present paper we will formulate a one-dimensional model of pulse
combustion; the model will be contrasted with earlier efforts in this
direction, and the resulting set of governing equations will be shown to
reduce, under an appropriate set of hypotheses, to the zero-space
dimensional case introduced in this section. The initial-boundary value
problem for the one-dimensional pulse combustor will also be compared to
other problems in the broad realm of gas dynamics which have been treated
extensively in the literature. In \cite{b3} we establish local and
global existence of smooth solutions for the nonlinear initial-boundary value
problem introduced in this paper. As the proof of local existence in \cite{b3} is
dependent on a fixed-point argument, we establish, in this paper, the
existence and uniqueness result for the relevant linearized
hyperbolic-parabolic system; this is accomplished by first regularizing this
system by introducing an artificial viscosity parameter $\delta$,
establishing existence and uniqueness for the resulting problem by using a
Galerkin argument, and then employing energy estimates, which are
independent of $\delta$, that allow us to let $\delta \to 0$ in the
regularized problem. The basic difficulty which must be overcome in both
this paper, as well as in \cite{b3}, is the influence of the time-dependent
boundary conditions associated with the influx of reactants into the pulse
combustion chamber.
\section{Previous efforts at including spatial effects in pulse combustion
modeling}
There have been a few attempts to develop a mathematical model of a pulse combustor which incorporates spatial dependence of the physical quantities but
there have been no known attempts to mathematically analyze the aforementioned models; in particular, the existence and uniqueness of solutions of the relevant
initial-boundary value problems associated with these models has not been addressed.
Moreover, the initial-boundary value problems associated with pulse combustion modeling differ from the majority of the gas-dynamics related initial boundary-value
problems in the literature; they are often defined on a bounded domain and lead to situations involving time-dependent boundary conditions.
In a pulse combustor reactants are added, and products are removed, periodically.
These properties are not unique to pulse combustor modeling; similar initial-boundary value problems arise in many other physical applications.
Therefore, the mathematical analysis presented in this paper may be of some significance for other physical problems as well.
We will now present a summary of the three mathematical models of pulse combustion referenced above.
Many processes in a pulse combustor are three-dimensional and are dominated by turbulent transport phenomena. However, since a typical pulse combustor system
has a large length to diameter ratio, the net influence of these processes
results in an unsteady, one-dimensional wave system. The flow field in a pulse
combustor can, over a large part of the combustor, be approximated by an oscillatory plug flow, thus, indicating that the flow can be simplified to be
one-dimensional.
In \cite{f1} a one-dimensional model was formulated and analyzed numerically;
the authors derive a coupled system of partial differential equations
following the standard procedures of continuum mechanics; i.e., they begin with balance
equations with a three-dimensional spatial dependence, namely,
\begin{subequations}\label{eq2.1}
\begin{gather}
\intertext{conservation of mass:}
\frac{\partial}{\partial t}\rho+\nabla\cdot(\rho {\bf v})=0
\intertext{conservation of momentum:}
\frac{\partial}{\partial t}(\rho{\bf v})+\nabla\cdot(\rho {\bf v}\otimes{\bf v} )= \nabla\cdot {\bf T}+\rho{\bf b}
\intertext{conservation of energy:}
\frac{\partial}{\partial t}(\rho\varepsilon)+\nabla\cdot(\rho\varepsilon {\bf v} )
= {\bf T}\cdot {\bf D}-\nabla \cdot{\bf h}+\rho\sigma
\intertext{and the entropy inequality}
\frac{\partial}{\partial t}(\rho\eta)+\nabla\cdot(\rho\eta {\bf v} )
\ge-\nabla \cdot({\bf h/\theta})+\rho\sigma/\theta
\end{gather}
\end{subequations}
where $\rho$ is the density, $\bf v$ is the velocity vector, $\varepsilon$
is the specific internal energy, $\bf h$ is the heat conduction vector,
$\bf T$ is the stress tensor, $\bf b$ is the specific body force vector,
$\bf D$ is the deformation rate tensor, $\theta$ is the temperature,
$\sigma$ is the
specific radiation, $\eta$ is the specific entropy, and $\otimes $
is the standard tensor product of vectors.
Assuming a one-dimensional dependence for all of the variables involved, and
introducing a heat conduction sink term $q_c^{\perp}$ to account for
heat loss in the direction orthogonal to the axis of symmetry,
the authors arrive at the system
\begin{subequations}\label{eq2.2}
\begin{gather}
\frac{\partial}{\partial t}(\rho A)+\frac{\partial}{\partial x}(\rho A v)=0
\\
\frac{\partial}{\partial t}(\rho Av)+\frac{\partial}{\partial x}(\rho Av^2 )
=\frac{\partial}{\partial x}(TA)
\\
\frac{\partial}{\partial t}(\rho A\varepsilon)
+\frac{\partial}{\partial x}(\rho A\varepsilon v )
= AT\frac{\partial v}{\partial x}-\frac{\partial}{\partial x}(A h)-q_c^{\perp}+\rho A\sigma+ A\dot{Q}
\end{gather}
\end{subequations}
where $x$ is the variable along the axis of the pulse combustion chamber
and the tailpipe
(see Figure \ref{fig1}),
$A$ is the cross sectional area of the pulse combustor, and $\dot{Q}$
is the heat generated by combustion.
As initial values the authors \cite{f1} assume atmospheric pressure, zero velocity,
and room temperature; i.e.,
\begin{subequations}
\begin{equation}\label{eq2.3a}
p(x,0)=p_0,\quad v(x, 0)=v_0, \quad \theta(x,0)=\theta_0
\end{equation}
The outlet of the tailpipe is considered to be the open end of an organ pipe; i.e., at
$x=L$ there is a pressure node and a maximum amplitude of the velocity:
\begin{equation}
\label{eq2.3b}
p(L,t)=p_0,\quad \frac{\partial v}{\partial x}(L, t)=0, \quad
\frac{\partial \rho}{\partial x}(L, t)=0,
\end{equation}
Fureby and Lundgren \cite{f1} also distinguish between closed and open valves
at the
entrance to the pulse combustor; this results in two sets of boundary conditions
at $x=0$.
In the case of a closed valve the pressure amplitude has a maximum and the mass
transport is zero; i.e.,
\begin{equation}\label{eq2.3c}
\frac{\partial p}{\partial x}(0,t)=0, \quad \{\rho Av\}(0,t)= 0,\quad
\frac{\partial\theta}{\partial x} (0,t)=0
\end{equation}
In case of an open valve, the temperature and pressure are assumed
to be the same as that in the gas supply line, and the mass transport
is modeled separately, specifically,
\begin{equation}\label{eq2.3d}
p(0,t)=p_g, \quad\{\rho Av\}(0,t)=\dot{m}(t)\ne 0,\quad \theta (0,t)=\theta_g
\end{equation}
\end{subequations}
To close the system of equations the following set of constitutive relations
is employed:
\begin{subequations}\label{eq2.4}
\begin{gather}
{\bf T}=(-p-\lambda tr {\bf D}){\bf I}+2\mu {\bf D}+\nu {\bf g} \otimes {\bf g},\\
{\bf h}=(k+\beta tr {\bf D}+\delta( tr {\bf D})^2 ){\bf g}+\gamma {\bf D}{\bf g},\\
\epsilon=\epsilon (\rho, \theta),\\
\eta=\eta (\rho, \theta)
\end{gather}
\end{subequations}
where ${\bf g}=\nabla\theta$ and $\lambda, \mu, \nu, \beta, \delta , \gamma$,
and $k$ are material constants, e.g., $\mu$ is the viscosity, and $k$ is the
thermal conductivity. It is, however, unclear as to what specific form of
these constitutive relations was used in the numerical experiments reported
in \cite{f1}.
In \cite{f1} submodels were also introduced to deal with the combustion processes,
e.g., for the energy release term $\dot{Q}$ it was assumed
that for some spatially varying amplitude function $K(x)$
\begin{equation}\label{eq2.5}
\dot{Q}=\dot{Q}(x,t)=K(x)\sin\Big(2\pi\frac{t}{\tau}-\delta\Big)
\end{equation}
where $\tau$ denotes the period and $\delta$ is the phase difference
between the mass flow $\dot{m}$ and the energy release $\dot{Q}$.
A heat transfer submodel of the form
\begin{equation}\label{eq2.6}
q_c^{\perp}=O(x)h_r(x,t)
\end{equation}
was incorporated into the model, where $O(x)$ is the circumference of the
combustion chamber at $x$, and $h_r(x,t)$ is the radial component of the
heat conduction vector $\bf h$. Finally, the following valve
model was also introduced:
\begin{equation}\label{eq2.7}
\dot{m}(t)=\{\rho Av\}(0,t)
=\frac{\partial}{\partial x}\int_t^{t+\tau}(TA)(x,\xi)d\xi\Big|_{x=0}
\end{equation}
where $\dot{m}(t)$ is the mass flow rate through the valve.
The other well-known contribution to the literature on one-dimensional pulse
combustion modeling may be found in \cite{b1} where unsteady, one-dimensional equations
of continuity, momentum, and energy were numerically solved; the model here also
allows for a variable area geometry, and assumes the perfect gas equation
of state.
The full model has the form
\begin{subequations}\label{eq2.8}
\begin{gather}
\frac{\partial (\rho A)}{\partial x} = -\frac{\partial}{\partial x} (\rho u A)\\
\frac{\partial (\rho u A)}{\partial x} = -\frac{\partial}{\partial x} (\rho u^2 A+pA) +p \frac{dA}{dx} - \rho A \frac{4f}{D} \frac{u^2}{2} \frac{u}{|u|}\\
\frac{\partial (\rho A E_s)}{\partial x} = -\frac{\partial}{\partial x} [u(\rho AE_s + pA)] +\dot{q} - 4Dh (T-T_{air})\\
E_s = c_VT + \frac{u^2}{2},\\
p = \rho RT
\end{gather}
\end{subequations}
Here $p$, $\rho$, and $T$ are, respectively, the pressure, density,
and temperature of the gas, $c_V$ is the specific heat at constant volume,
$u$ is the fluid velocity, $D$ and $A$ are the local side and cross-sectional
area of the square combustor, $T_{air}$ is the external temperature used to
determine heat losses, $f$ is the friction factor, and $\dot{q}$ is the
heat generated due to the combustion process.
The authors \cite{b1} use the following initial and boundary conditions in
their numerical computations:
\begin{subequations}
\begin{equation}\label{eq2.9a}
p(x,0)=p_{\rm atm}, \quad T(x,0)=T_{air},\quad u(x,0)=0
\end{equation}
During injection, at the entrance, it was assumed that
\begin{equation}\label{eq2.9b}
\frac{\partial p}{\partial x}(0,t)=0, \quad T(0,t)=T_{air},\quad
u(0,t)=\frac{\dot{m}(t)}{\rho A}
\end{equation}
and for the case where valve is closed the boundary conditions were
\begin{equation}\label{eq2.9c}
\frac{\partial p}{\partial x}(0,t)=0, \quad
\frac{\partial T}{\partial x}(0,t)=0,\quad u(0,t)=0
\end{equation}
Finally, at the tailpipe exit, it was assumed that
\begin{equation}\label{eq2.9d}
p(L,t)=p_{\rm atm}, \quad \frac{\partial \rho}{\partial x}(L,t)=0,\quad
\frac{\partial u}{\partial x}(L,t)=0
\end{equation}
\end{subequations}
We note that, mathematically, only five boundary conditions are required
in each of the models governed by systems \eqref{eq2.2} and \eqref{eq2.8}.
Specifying six boundary conditions makes these systems mathematically
overdetermined; however, all six boundary conditions are necessary for a numerical
treatment of the problems specified above and, indeed, these papers
only employ numerical treatments of the models.
It is not surprising that some of the boundary conditions are slightly different
in the models developed in \cite{b1,f1}.
Indeed, in both \cite{b1} and \cite{f1} a version of the McCormack predictor-corrector method is used to solve the system of partial differential equations subject to boundary
and initial conditions; this method is mathematically equivalent to introducing
an artificial viscosity parameter and then using standard numerical methods for
parabolic equations. In each case the special pulse combustor data introduced
for each model was used for calculations and many of the parameters were chosen
specific to the particular pulse combustor. While \cite{f1} simply checks the
consistency of the model with experiments, the authors of \cite{b1} also attempted
to find an optimal frequency of operation for the pulse combustor.
\section{A one-dimensional, lumped parameter, pulse combustion model}
The general form of the equations describing the motion of a reactive
gas is based on the following conservation laws:
\begin{itemize}
\item[(i)] conservation of mass:
\begin{equation}\label{eq3.1}
\frac{\partial }{\partial t}\left(\rho A\right)
=-\frac{\partial }{\partial x}(\rho u A)
\end{equation}
\item[(ii)] conservation of momentum:
\begin{equation}\label{eq3.2}
\frac{\partial }{\partial t}\left(\rho uA\right)=-\frac{\partial }{\partial x}\left(\rho u^2 A-\sigma A\right)
\end{equation}
\noindent{and}
\item[(iii)] conservation of energy:
\begin{equation}\label{eq3.3}
\frac{\partial }{\partial t}(\rho AE)
=-\frac{\partial }{\partial x}\left(u\rho AE+HA-u\sigma A\right)+\dot{q}
\end{equation}
\end{itemize}
where, as in the previous section, $\rho(x,t)$ is the density of the gas,
$u(x,t)$ is the velocity, $T(x,t)$ is the gas temperature, $A(x)$ is the
cross sectional area of the pulse combustor, $\sigma(x,t)$ is the stress
tensor, $E(x,t)$ is energy per unit mass, $H(x,t)$ is heat conduction
in the axial direction, and $\dot{q}(x,t)$ is the heat released due to
chemical reactions per unit time.
The system
of equations \eqref{eq3.1}-\eqref{eq3.3} is consistent with the system in
\cite{f1} but there seems to be some inconsistency with the system
in \cite{b1}. In particular the conservation of momentum
equation \eqref{eq2.8}b has a form which seems to be inconsistent
with the principles of continuum mechanics.
The system of conservation equations \eqref{eq3.1}-\eqref{eq3.3}
are closed by the constitutive relations:
\begin{gather}\label{eq3.4}
E=c_VT+\frac{u^2}{2}, \\
\label{eq3.5}
p=\rho RT, \\
\label{eq3.6}
\sigma=-p+\mu\frac{\partial u}{\partial x},\\
\label{eq3.7}
H=-k\frac{\partial T}{\partial x}
\end{gather}
where $c_V$ is specific heat of the gas, $R$ is the gas constant, $\mu$ is the
gas viscosity, assumed to be constant, and $k$ is the heat conduction
coefficient, also assumed to be constant. This specific form of the constitutive
relations is consistent with the forms proposed in \cite{b1,f1}.
We choose as variables the density, velocity and temperature; all other
functions will be assumed to be functions of $\rho$, $u$, and $T$, with the
specific dependence expressed through the constitutive relations.
Using the constitutive relations, we obtain from \eqref{eq3.1}-\eqref{eq3.3}
the following evolution equations
\begin{gather}\label{eq3.8}
A\frac{\partial \rho}{\partial t}
=-Au\frac{\partial \rho}{\partial x}-A\rho\frac{\partial u }{\partial x}
-\frac{\partial A}{\partial x}(\rho u)
\\
\label{eq3.9}
\begin{aligned}
&Au \frac{\partial \rho}{\partial t} + A \rho \frac{\partial u}{\partial t}\\
&= -\frac{\partial \rho}{\partial x} u^2 A - \frac{\partial u}{\partial x} 2
\rho uA - \frac{\partial (\rho RT)}{\partial x} A + \mu \frac{\partial^2
u}{\partial x^2} A - \frac{\partial A}{\partial x} \Big( \rho u^2 - \rho RT
+ \mu \frac{\partial u}{\partial x} \Big)
\end{aligned}\\
\label{eq3.10}
\begin{aligned}
&A\rho c_V\frac{\partial T}{\partial t}
+ A\rho u\frac{\partial u }{\partial t}
+A\Big(c_VT+\frac{u^2}{2}\Big)\frac{\partial \rho}{\partial t}\\
&=-Auc_V\rho\frac{\partial T }{\partial x}-Au^2\rho
\frac{\partial u}{\partial x}
-Au\Big(c_VT+\frac{u^2}{2}\Big)\frac{\partial \rho}{\partial x}\\
&\quad -A\rho \Big(c_VT+\frac{u^2}{2}\Big)\frac{\partial u }{\partial x}
-A\frac{\partial }{\partial x}\Big( -\rho RT\frac{\partial u}{\partial x}
+\mu\big(\frac{\partial u}{\partial x}\big)^2\Big)+\dot{q}
\end{aligned}
\end{gather}
Using \eqref{eq3.8} to substitute for $ A\frac{\partial\rho}{\partial t}$
in \eqref{eq3.9} we obtain
\begin{equation}\label{eq3.11}
\begin{aligned}
&u\Big(-Au\frac{\partial \rho}{\partial x}
-A\rho\frac{\partial u }{\partial x}
-\frac{\partial A}{\partial x}(\rho u)\Big)
+ A\rho\frac{\partial u }{\partial t}\\
&=-\frac{\partial \rho }{\partial x} u^2 A-\frac{\partial u }{\partial x}
2\rho u A-\frac{\partial p}{\partial x}A+\mu\frac{\partial^2 u}{\partial x^2}A
-\frac{\partial A }{\partial x}\Big(\rho u^2 -p
+\mu\frac{\partial u}{\partial x}\Big)
\end{aligned}
\end{equation}
which can then be reduced to
\begin{equation}\label{eq3.12}
A\rho\frac{\partial u }{\partial t}
=-\frac{\partial u }{\partial x}\rho u A
-\Big(R\rho\frac{\partial T}{\partial x}
+RT\frac{\partial\rho}{\partial x}\Big)A
+\mu\frac{\partial^2 u}{\partial x^2}A
-\frac{\partial A }{\partial x}
\Big( -\rho RT+\mu\frac{\partial u}{\partial x}\Big)\,.
\end{equation}
We now use \eqref{eq3.8} and \eqref{eq3.12} to substitute for
$ A\frac{\partial\rho}{\partial t}$ and $ A\rho
\frac{\partial u}{\partial t}$, respectively, on the right hand side of
\eqref{eq3.10}. Assuming $\rho(x,t)>0$, we then obtain the
following system of three partial differential equations for $\rho$, $u$,
and $T$:
\begin{subequations}\label{eq3.13}
\begin{gather}
\frac{\partial \rho}{\partial t}+u\frac{\partial \rho}{\partial x}
+\rho \frac {\partial u}{\partial x}
=-\rho u \frac{\partial}{\partial x}(\ln A),
\\
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}
+R\frac{\partial T}{\partial x}+\frac{RT}{\rho}\frac{\partial \rho}{\partial x}
=\frac{\mu}{\rho}\frac{\partial^2 u}{\partial x^2}
+\frac{\mu}{\rho}\frac{\partial u}{\partial x}\frac{\partial }{\partial x}
(\ln A)-RT\frac{\partial }{\partial x}(\ln A),
\\
\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}
+\frac{RT}{c_V}\frac{\partial u}{\partial x}
=\frac{\mu}{c_V}\frac{1}{\rho}
\left(\frac{\partial u}{\partial{x}}\right)^2+\frac{k}{c_V}\frac{1}{\rho}
\frac{\partial^2T}{\partial x^2}+\frac{\dot{q}}{A c_V}
\frac{1}{\rho}+\frac{k}{c_V\rho}\frac{\partial T}{\partial x}
\frac{\partial}{\partial x}(\ln A)\,.
\end{gather}
\end{subequations}
Using the assumptions introduced in \cite{b3} it will be shown
that the vacuum state does not occur for initial data chosen sufficiently
small; this will justify the assumption that $\rho > 0$ in
\eqref{eq3.13}.
The valves at the entrance to the pulse combustor are assumed to be designed in
such a way as to allow control over the reactant gas flow into the chamber.
We assume, therefore, that the velocity of the gas entering the pulse combustor
chamber is a known function of time; i.e., at $x=0$ we have
\begin{equation}\label{eq3.14}
u(0,t)=\alpha(t)
\end{equation}
for some function $\alpha(t)$. The model does not differentiate between open and
closed valves. We assume, instead that, due to valve inertia the valve never
closes completely, which seems to be a physically realistic assumption; this
allows for a continuous (although oscillating) flow of the reactants into the
chamber. We impose the following conditions on the function $\alpha$:
\begin{gather}\label{eq3.15}
\alpha(t)\ge \alpha_0>0,\quad t\ge0,\\
\label{eq3.16}
\alpha(t)\in C^3(0,\infty)\,.
\end{gather}
The first condition reflects the fact that the flow is always directed towards
the chamber and the valves never close completely. The second condition
guarantees that the function $\alpha(t)$ is sufficiently smooth for the analysis
to follow.
We also assume that we can control the temperature and pressure and, hence, the
density of the reactants flowing into the chamber; this assumption is equivalent
to the following entrance boundary conditions on $\rho$ and $T$.
\begin{gather}\label{eq3.17}
T(0,t)=T_{\rm in}>0,\\
\label{eq3.18}
\rho(0,t)=\rho_{\rm in}=\frac{p_{\rm atm}}{RT_{\rm in}}>0\,.
\end{gather}
\noindent \textbf{Remarks:} The current model does not allow for flow reversal in the
tailpipe. In order to incorporate flow reversal, the mathematical domain of the
problem would need to be extended beyond the exit of the tailpipe, and some
mixing mechanism would need to be introduced, likely requiring a model with
at least a two dimensional spatial dependence.
As the system \eqref{eq3.13}a,b,c is first order $\rho$, no exit boundary condition
can be imposed with respect to $\rho$ at $x=L$; introducing such a boundary
condition will result in an over determined system. The conditions for $u$
and $T$ at $x=L$ are the following: first of all,
the flow of the gas exiting the tailpipe is incompressible; i.e.,
\begin{equation}\label{eq3.19}
\frac{\partial u}{\partial x}(L,t)=0
\end{equation}
Next, once a stable operating cycle of the pulse combustor has been established,
the temperature of the combustion products coming out of the tailpipe remains
constant, as it depends solely on the air-fuel ratio used. Therefore, we
seek solutions for which
\begin{equation}\label{eq3.20}
T(L,t)=T_{\rm out}={\rm const.}
\end{equation}
This latter condition may also be substantiated as follows:
the set of exit boundary conditions in the previous formulations of
one-dimensional pulse combustion in \cite{b1,f1}; i.e.,
$$
\rho_x(L,t)=0, \quad
u_x(L,t)=0, \quad
p(L,t)=p_{\rm atm}
$$
yield, as has been noted, an overdetermined problem and does not contain
a boundary condition for the temperature. However, for the exit boundary
conditions in the current
model \eqref{eq3.19}, \eqref{eq3.20}, the boundary condition for
the temperature follows as a consequence of the boundary conditions for the
density and the pressure which were used in \cite{b1,f1}.
In fact, using the conservation of mass equation we obtain
\begin{equation}\label{eq3.21}
\rho_t(L,t)+u(L,t)\rho_x(L,t)+\rho(L,t) u_x(L,t)
=-\rho(L,t) u(L,t) \frac{ A_x(L)}{A(L)}
\end{equation}
If we then apply \eqref{eq3.19}, and the fact that the cross-section has
constant area along the length of the tailpipe, we obtain
$\rho_t (L, t) = 0$
which, when combined with \eqref{eq3.19}, implies that
\begin{equation}\label{eq3.22}
\rho(L,t)=\textrm{const}
\end{equation}
As a consequence of the ideal gas law, however,
$$
T(L,t)=\frac{p_{\rm atm}}{R\rho(L,t)}={\rm const.}
$$
which is \eqref{eq3.20}.
For the initial conditions at $t=0$ we assume the specification of
sufficiently smooth functions of $x$; i.e.,
\begin{subequations}\label{eq3.23}
\begin{gather}
u(x,0)=u_0(x) \\
\rho(x,0)=\rho_0(x) \\
T(x,0)=T_0(x)
\end{gather}
\end{subequations}
where $u_0(x)$, $\rho_0(x)$, $T_0(x)\in C^2[0,L]$.
The complete model considered in this paper, as well as in \cite{b3},
consists of the system of equations \eqref{eq3.13}a,b,c for $\rho, u, T$,
the boundary conditions \eqref{eq3.14}, \eqref{eq3.17}, \eqref{eq3.18},
\eqref{eq3.19}, and \eqref{eq3.20}, and the initial data \eqref{eq3.23}a,b,c,
a sketch of the pulse combustor configuration associated with this model is
presented in Figure 3.
\section{Some related work on problems in gas dynamics}
Initial-boundary value problems associated with pulse combustion
modeling differ from the majority of the gas-dynamics related initial
boundary-value problems in the literature; such problems are often defined
on a bounded domain and lead to situations involving time-dependent boundary
conditions. In a pulse combustor reactants are added, and products are
removed, periodically. These properties are not unique to pulse combustor
modeling; similar initial-boundary value problems arise in other physical
applications, e.g., blood flow \cite{c1} and the references contained therein.
Existence and uniqueness for initial and initial-boundary value problems
associated with the motion of viscous, compressible fluids has been covered
extensively in the literature
\cite{i1,k1,k2,k4,k6,l2,m2,m3,n1,n2,n3,s1,s3,t1,v1,v2}.
This includes work related to the gas dynamics equations with a
three-dimensional spatial dependence \cite{i1,m2,n2,s1,t1}.
In \cite{m2}, the equations of motion of compressible viscous and
heat-conductive fluids were investigated for initial boundary value problems
in a half space
and in the exterior domain of any bounded region. A globally unique solution
(in time) was proved to exist and approach the stationary state as
$t\to \infty$, provided the prescribed initial data and the external
force were sufficiently small.
The solutions, in fact, possess the following smoothness:
\begin{equation}\label{eq4.1}
\begin{gathered}
\rho\in C^0(0,\infty; H^3(\Omega))\cap C^1(0,\infty; H^2(\Omega)),\\
u, \theta\in C^0(0,\infty; H^3(\Omega))\cap C^1(0,\infty; H^1(\Omega))
\end{gathered}
\end{equation}
Tani \cite{t1} establishes existence and uniqueness results for the first
initial-boundary value problem of compressible viscous fluid motion,
and Itaya \cite{i1} provides a similar result for the Cauchy problem.
The system of gas dynamics equations with a two-dimensional spatial dependence
was considered, for example, by Kazhikhov and Vaigant \cite{v1}.
In particular the existence of a unique solution
\begin{equation}\label{eq4.2}
u(x,y,t)\in C^{2+\alpha, 1+\alpha/2}(\bar{Q}_T), \quad
\rho(x,y,t)\in C^{1+\alpha, 1+\alpha/2}(\bar{Q}_T)
\end{equation}
($Q_T=\Omega\times[0,T]$, $\Omega$ being the spatial domain)
was established, provided the initial data were sufficiently small
and satisfied
\begin{equation}\label{eq4.3}
u^0(x,y)\in C^{2+\alpha}(\bar{\Omega}), \quad
\rho^0(x,y)\in C^{1+\alpha}(\bar{\Omega})
\end{equation}
In \cite{k4}, the existence of global classical solutions to initial
boundary value problems in the dynamics of a one-dimensional, viscous,
heat-conducting gas was established; in this work nonlinear dissipative
effects turn out to be sufficiently strong to prevent the development of
singularities. In \cite{k6}, a system of equations for a viscous
heat-conducting perfect gas was studied for the case of a one-dimensional
motion with plane waves; the unique solvability of the problem of gas flow in a
bounded region with impermeable thermally insulated boundaries was proven for
the class of the both generalized (strong) and classical solutions.
An existence theorem was established by using a priori estimates to extended
the local (in time) solution to a global solution; a major role is played here
by upper and lower bounds for the density and temperature. Kanel \cite{k1}
provided an existence result for the Cauchy problem with one-dimensional
spatial dependence. Finally, the review paper by Solonnikov and Kazhikhin
\cite{s3} provides a good survey of existence results for the one-dimensional
equations governing the motion of a compressible fluid.
All of the existence results cited above are, however, valid only for
homogeneous systems of partial differential equations with homogeneous
boundary conditions. For the case of the one-dimensional equations governing
the motion of a compressible fluid, a change of variables from density to
specific volume can be introduced (Lagrangian variables). This change of
variables significantly simplifies the conservation equations; in particular,
the conservation of mass equation reduces to $v_{,t} - u_{,x} =0$,
where $v=1/\rho$ is the specific volume of the gas. However, this change of variables
assumes that the density remains strictly positive; the possibility of a
vacuum state must be addressed separately \cite{s3}. The Lagrangian change
of variables can not be applied to the initial-boundary value problem
formulated in section 3 as the problem introduced there does not involve
a homogeneous system of equations, nor does it possess homogeneous
boundary conditions. The analysis presented in this paper, as well as in
\cite{b3}, is closest in spirit to that of the energy argument employed in \cite{s2},
albeit without the type of time-dependent boundary conditions which appear
in the present work.
\section{Relation of the current model to the spatially independent model}
If one assumes spatial independence, as well as the other assumptions of the
model described in section 3, we can expect the one-dimensional model
to reduce to the spatially independent model described in \cite{b6}.
In this section we will demonstrate that this, in fact, is the case.
The one-dimensional model is governed by the system of three
conservation laws which result from combining \eqref{eq3.1}-\eqref{eq3.3}
with \eqref{eq3.6}, assuming $\mu=0$; this procedure yields the system
\begin{gather}\label{eq5.1}
\frac{\partial }{\partial t}\left(\rho A\right)
=-\frac{\partial }{\partial x}(\rho u A),\\
\label{eq5.2}
\frac{\partial }{\partial t}\left(\rho uA\right)
=-\frac{\partial }{\partial x}\left(\rho u^2 A+pA\right), \\
\label{eq5.3}
\frac{\partial }{\partial t}(\rho AE)
=-\frac{\partial }{\partial x}\left[u\left(\rho AE+pA\right)\right]+\dot{q}
\end{gather}
We also append the constitutive relations \eqref{eq3.4}, \eqref{eq3.5} for the
total energy and the pressure.
The spatially independent model involves two equations. The first one
is a consequence of energy balance in the combustion chamber and is the
precursor to \eqref{eq1.1}, namely,
\begin{equation}\label{eq5.4}
\frac{d }{d t}\left[\rho_Re_RV_R+\rho _Pe_PV_P\right]
=h_R\dot{m_R}+\frac{\Delta H}{1+r}\dot{m}_B-h_0\dot{m_0}
\end{equation}
where $\rho_R$ is the density of the reactants, $\rho_P$ is the density of the
combustion products, $e_R$ and $e_P$ are the internal energy of the reactants and
products, respectively, (per unit mass of the mixture), $V_R$($V_P$) is the
chamber volume occupied by reactants (products), $\dot{m}_R$ is the mass flow
rate of the reactants, $\dot{m}_B$ is the mass burning rate of the reactant
mixture, $\dot{m}_0$ is the mass flow rate of the combustion products in the
tailpipe, $r$ is the air fuel (mass) ratio, $\Delta H$ is the heat of combustion
per unit mass of the fuel, $h_R$ is the enthalpy, per unit mass of
the reactant mixture entering the combustion chamber, and $h_0$ is the
enthalpy, per unit mass of the mixture of combustion products leaving the
chamber. We note that $V_B=V_R+V_P$, where $V_B$ is the combustion chamber
volume. The second equation in the model is the momentum equation in the tailpipe
\eqref{eq1.2}, which we repeat here as
\begin{equation}\label{eq5.5}
\bar{\rho} L\frac{du }{ dt}=p-p_a
\end{equation}
with $\bar{\rho}$ the average density of the combustion products in the
tailpipe, $p$ the pressure in the tailpipe, $p_a$ the atmospheric pressure,
$L$ the length of the tailpipe, and $u$ the velocity of the gas in the tailpipe.
We first consider the energy equation \eqref{eq5.3} and
recall the following three relationships from thermodynamics
\begin{equation}
\label{eq5.6}
c_P-c_V=R
\end{equation}
where $c_P$ is the specific heat at constant pressure, per unit mass of the gas,
\begin{equation}\label{eq5.7}
h=c_PT
\end{equation}
where $h$ is the enthalpy of the gas, per unit mass, and
\begin{equation}
\label{eq5.8}
e=c_VT
\end{equation}
where $e$ is internal energy of the gas, per unit mass.
In the spatially independent case the velocity inside the chamber is zero;
therefore, in the chamber
\begin{equation}
\label{eq5.9}
E=c_VT+\frac{u^2}{2}=e
\end{equation}
which can also be rewritten using \eqref{eq5.7} and the ideal gas law
\eqref{eq3.5} as
\begin{equation}
\label{eq5.10}
E=c_VT=(c_P-R)T=h-RT=h-\frac{p}{\rho}
\end{equation}
Substituting for $E$ in \eqref{eq5.3} we obtain
\begin{equation}
\label{eq5.11}
\frac{\partial }{\partial t}(\rho Ae)=-\frac{\partial }{\partial x}
\Big[u\Big(\rho A(h-\frac{p}{\rho})+pA\Big)\Big]+\dot{q}
\end{equation}
which yields
\begin{equation}\label{eq5.12}
\frac{\partial }{\partial t}(\rho Ae)=-\frac{\partial }{\partial x}\left(\rho u Ah\right)+\dot{q}
\end{equation}
We now integrate equation \eqref{eq5.12} along the length of the chamber with
the entrance of the chamber at $x=0$ and the exit of the chamber at $x=l$:
\begin{equation}\label{eq5.13}
\int_0^{l} \frac{\partial}{\partial t} (\rho Ae)dx =
-\int_0^{l} \frac{\partial}{\partial x} (\rho u Ah)dx +
\int_0^{l} \dot{q}dx
\end{equation}
\noindent
\textbf{Remarks:} In the model introduced in section 3, the entrance to the
combustion chamber is at $x=0$ and the exit from the tailpipe is at $x=L$;
here we assume that the combustion chamber occupies the domain $0 \leq x
\leq l$ and the tailpipe the domain $l < x \leq
l + L$.
We now note that
\begin{equation}
\label{eq5.14}
e=c_VT=c_{VR}T\nu_R+c_{VP}T\nu_P
=c_{VR}T\frac{\rho_R V_R}{\rho V}+c_{VP}T\frac{\rho_P V_P}{\rho V}
\end{equation}
where $\nu_R$ and $\nu_P$ are mass fractions of the reactants and products,
respectively. Since $V=Al$ we can write
\begin{equation} \label{eq5.15}
\rho A e=c_{VR}T\frac{\rho_R V_R}{l}+c_{VP}T\frac{\rho_P V_P}{l}
=\frac{1}{l}(e_R\rho_RV_R+e_P\rho_PV_P)
\end{equation}
We have, from \eqref{eq5.13},
\begin{equation}
\label{eq5.16}
\frac{\partial}{\partial t} \int_0^{l} \frac{1}{l}
(e_R \rho_R V_R+e_P\rho_PV_P)dx=(\rho uAh)\mid_0^{l} +
\int_0^{l} \dot{q}dx
\end{equation}
which yields
\begin{equation}\label{eq5.17}
\frac{d}{dt}(e_R\rho_RV_R+e_P\rho_PV_P)=(\rho uA)_{\rm in}h _{\rm in}-(\rho uA)_{\rm out}h_{\rm out}+\dot{q}_{total}
\end{equation}
where
\begin{equation}
\label{eq5.18}
\dot{q}_{total}=\int_0^{l} \dot{q}dx
\end{equation}
is the total heat released in the chamber due to chemical reactions, per unit
mass of the gas mixture, per unit time. In the spatially independent model,
combustion of the fuel is completely achieved and hence
\begin{equation}\label{eq5.19}
\dot{q}_{total}=\frac{\Delta H}{1+r}\dot{m}_B
\end{equation}
Also, because of the assumptions in the spatially independent model we have
$(\rho uA)_{\rm in}=\dot{m}_R$, $(\rho uA)_{\rm out}=\dot{m}_0$, $h_{\rm in}=h_R$ and
$h_{\rm out}=h_0$. Substituting these assumptions into \eqref{eq5.17} we obtain
the zero-dimensional energy equation \eqref{eq5.4}.
To arrive at the spatially-independent momentum equation \eqref{eq5.5}
we consider the momentum equation \eqref{eq5.2} in the tailpipe. We integrate
\eqref{eq5.2} along the length of the tailpipe so as to obtain
\begin{equation}
\label{eq5.20}
\frac{\partial }{\partial t}\int_l^{l+L}(\rho uA)dx=-\int_l^{l+L}\frac{\partial }{\partial x}\left(\rho u^2A+pA\right)dx
\end{equation}
where we used the fact that $\frac{\partial A}{\partial x}=0$ in the spatially
independent case. In fact, as $A$ is constant in the tailpipe, while the
velocity $u$ of the gas is spatially constant, we obtain
\begin{equation}
\label{eq5.21}
A\frac{\partial }{\partial t}\left(u \int_l^{l+L}\rho dx\right)
=-\left((\rho uA)u+pA\right)\mid_l^{l+L}
\end{equation}
However, the mass flow rate $\rho uA=\dot{m}_0$ is constant along the length
of the tailpipe, therefore, with $\bar{\rho}$ the average density in the
tailpipe, it follows from \eqref{eq5.21} that
\begin{equation}
\label{eq5.22}
A\bar{\rho}L\frac{du}{dt}=A(p(l)-p(L+l))
\end{equation}
which is, of course, equivalent to \eqref{eq5.5}.
\section{Existence and uniqueness for the linear system
with artificial viscosity}
The initial-boundary value problem for the one-dimensional pulse combustion
model introduced in \S3 consists of the mixed hyperbolic-parabolic system
\eqref{eq3.13} a,b,c, the boundary conditions \eqref{eq3.14}, \eqref{eq3.15},
\eqref{eq3.16}, \eqref{eq3.19}, and \eqref{eq3.20}, and the initial data
\eqref{eq3.23} a,b,c.
We begin the analysis in this section by effecting a change of variables so as
to obtain a problem with homogeneous boundary conditions; more specifically,
we set
\begin{subequations}\label{eq6.1}
\begin{gather}
\hat{u}=u-\alpha(t) \\
\hat{\rho}=\rho -\rho_{\rm in} \\
\hat{T}=T -\frac{x}{L}T_{\rm out} -\frac{L-x}{L}T_{\rm in}
\end{gather}
\end{subequations}
and substitute \eqref{eq6.1}a,b,c in \eqref{eq3.13}a,b,c so as to obtain,
after rearranging terms, the system
\begin{subequations}\label{eq6.2}
\begin{align}
&\hat{\rho}_t+a(x,t)\hat{u}_x+b(x,t)\hat{\rho}_x=c(x,t)\\
\hat{u}_t+b_2(x,t)&\hat{u}_x+R\hat{T}_x+d(x,t)\hat{\rho}_x=f(x,t)\hat{u}_{xx}+g(x,t)\\
\hat{T}_t&+b_3(x,t)\hat{T}_x+h(x,t)\hat{u}_x=i(x,t)\hat{T}_{xx}+j(x,t)
\end{align}
\end{subequations}
The coefficients in \eqref{eq6.2}a,b,c are given, explicitly, by
\begin{subequations}\label{eq6.3}
\begin{gather}
a(x,t)=\hat{\rho}+\rho_{\rm in}\\
b(x,t)=\hat{u}+\alpha(t)\\
b_2(x,t)=\hat{u}+\alpha(t)-\frac{\mu}{\hat{\rho}+\rho_{\rm in}}(\ln A)_x\\
b_3(x,t)=\hat{u}+\alpha(t)-\frac{k}{c_V(\hat{\rho}+\rho_{\rm in})}(\ln A)_x\\
c(x,t)=-(\hat{\rho}+\rho_{\rm in})(\hat{u}+\alpha(t))(\ln A)_x\\
d(x,t)=\frac{R}{\hat{\rho}+\rho_{\rm in}}
\Big(\hat{T} +\frac{x}{L}T_{\rm out} +\frac{L-x}{L}T_{\rm in}\Big)\\
f(x,t)=\frac{\mu}{\hat{\rho}+\rho_{\rm in}}\\
g(x,t)=-\alpha'(t)-\frac{R}{L}(T_{\rm out}-T_{\rm in})
-R\Big(\hat{T} +\frac{x}{L}T_{\rm out}
+\frac{L-x}{L}T_{\rm in}\Big)(\ln A)_x\\
h(x,t)=\frac{\mu}{c_V}\frac{ \hat{u}_x}{\hat{\rho}+\rho_{\rm in}}+\frac{R}{c_V}
\Big(\hat{T} +\frac{x}{L}T_{\rm out} +\frac{L-x}{L}T_{\rm in}\Big)\\
i(x,t)=\frac{k}{c_V(\hat{\rho}+\rho_{\rm in})}\\
j(x,t)=-(\hat{u}+\alpha(t))\Big(\frac{T_{\rm out}-T_{\rm in}}{L}\Big)
+\frac{\dot{q}}{c_V(\hat{\rho}+\rho_{\rm in})}
\end{gather}
\end{subequations}
We note that $h(x,t)$ is the only coefficient containing a derivative of one of
the unknown functions. Also, in view of \eqref{eq6.1}a, b,c, the boundary
data and initial conditions assume the following form:
For the boundary conditions at $x=0$ we have, for all $t>0$,
\begin{subequations}\label{eq6.4}
\begin{gather}
\hat{u}(0,t)=u(0,t)-\alpha(t)=0\\
\hat{\rho}(0,t)= \rho(0,t) -\rho_{\rm in}=0\\
\hat{T}(0,t)=T(0,t) -T_{\rm in}=0\\
\intertext{while those at $x=L$ assume the form}
\hat{u}_x(L,t)=0\\
\hat{T}(L,t)=T(L,t) -T_{\rm out}=0
\end{gather}
\end{subequations}
In terms of the new variables, the initial conditions are
\begin{subequations}\label{eq6.5}
\begin{gather}
\hat{u}(x, 0) = u_0(x) - \alpha(0) = \hat{u}_0 (x) \\
\hat{\rho} (x,0) = \rho_0 (x) - \rho_{\rm in} = \hat{\rho}_0(x) \\
\hat{T}(x,0) = T_0(x) - \frac{x}{L} T_{\rm out} - \frac{L-x}{L} T_{\rm in} =
\hat{T}_0 (x)
\end{gather}
\end{subequations}
If we ignore the dependence of the coefficients in \eqref{eq6.3}a-k on
$\hat{\rho}$, $\hat{u}$, $\hat{T}$ and instead assume that the coefficients
$a$, $b$, $b_2$, $b_3$, $c$, $d$, $f$, $g$, $h$, $i$, $j$ are known functions
of $x$ and $t$ only, then by dropping the hats on $\hat{\rho}$, $\hat{u}$, and
$\hat{T}$, including those on $\hat{\rho}_0$, $\hat{u}_0$, and $\hat{T}_0$,
we obtain the linear initial-boundary value problem
\begin{subequations}\label{eq6.6}
\begin{gather}
\rho_t+au_x+b\rho_x=c\\
u_t+b_2u_x+RT_x+d\rho_x=fu_{xx}+g\\
T_t+b_3T_x+hu_x=iT_{xx}+j
\end{gather}
\end{subequations}
with initial data on $[0,L]$
\begin{subequations}\label{eq6.7}
\begin{gather}
u(x,0)=u_0(x)\\
\rho(x,0)=\rho_0(x)\\
T(x,0)=T_0(x)
\end{gather}
\end{subequations}
and, for all $t >0$, the boundary data
\begin{subequations}\label{eq6.8}
\begin{gather}
\rho(0,t)=0\\
u(0,t)=0\\
T(0,t)=0\\
u_x(L,t)=0\\
T(L,t)=0
\end{gather}
\end{subequations}
The purpose of this paper is to prove an existence and uniqueness theorem
for the system \eqref{eq6.6}, \eqref{eq6.7}, \eqref{eq6.8} (as well as for a
regularized version of this system); the latter result will
serve as the starting point for the local and global existence results
for the original nonlinear problem \eqref{eq3.13}a,b,c, \eqref{eq3.14},
\eqref{eq3.17}-\eqref{eq3.20}, and \eqref{eq3.23}a,b,c in \cite{b3}. In fact, the
existence and uniqueness result for \eqref{eq6.6}, \eqref{eq6.7},
\eqref{eq6.8}, which is established in \S7, depends on proving, in this
section, a related result for the regularized version of this system which is
introduced below; this regularized problem is treated by using the method of
Galerkin approximations coupled with an energy argument. For the problem
\eqref{eq6.6}, \eqref{eq6.7}, \eqref{eq6.8}, as well as for the regularized
version of this problem possessing an artificial viscosity, we will assume
that
\begin{subequations}\label{eq6.9}
\begin{gather}
f(x,t)\ge f_c>0, \quad \forall(x,t)\in [0,t]\times[0,L]\\
i(x,t)\ge i_c>0, \quad \forall(x,t)\in [0,t]\times[0,L]\\
a, b, b_2, b_3, c, d, f, g, h, i, j
\in C^\infty([0,t]; C^\infty[0,L])
\end{gather}
\end{subequations}
To regularize the mixed, linear, hyperbolic-parabolic system \eqref{eq6.6}
we add the viscous term $\delta \rho_{xx}$ to \eqref{eq6.6}a where $\delta
>0$ is an artificial viscosity; this produces the uniformly parabolic system
\begin{subequations}\label{eq6.10}
\begin{gather}
\rho_t+au_x+b\rho_x=\delta\rho_{xx}+c\\
u_t+b_2u_x+RT_x+d\rho_x=fu_{xx}+g\\
T_t+b_3T_x+hu_x=iT_{xx}+j\,.
\end{gather}
\end{subequations}
For the system \eqref{eq6.10}a,b,c we retain the initial data
\eqref{eq6.7}a,b,c but, as we have increased the
order of the equation governing the evolution of $\rho$, we append to the
boundary data \eqref{eq6.8} the additional boundary condition
\[
\rho_x (L, t) = 0, \quad t > 0
\eqno{(6.8f)}
\]
The full regularized problem now consists of \eqref{eq6.10}a,b,c,
\eqref{eq6.7}a,b,c, and \eqref{eq6.8}a-f; to deal with this problem we begin
by introducing the spaces which are used in the Galerkin approximations;
i.e., we have the following definition.
\begin{definition}\label{def:6.1}
For $m\in \mathbb{N}$ define the finite dimensional spaces
\begin{subequations}\label{eq6.11}
\begin{gather}
\mathcal{V}_m=\big\{v(x,t): v=\sum_{l=0}^m
\alpha_l(t)\sqrt\frac{2}{{L}}\sin\frac {(2l+1)\pi x}{2L}\big\},,\\
\mathcal{W}_m=\big\{w(x,t): w=\sum_{l=0}^m \beta_l(t)
\sqrt\frac{2}{{L}}\sin\frac {l\pi x}{L}\big\}\,,
\end{gather}
\end{subequations}
where $\alpha_l(\tau)$, $\beta_l(\tau)\in C^1[0,t]$.
\end{definition}
Recalling that the functions $\rho, u, T$ which appear in
\eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c, and \eqref{eq6.8}a-f are, in fact,
the $\hat{\rho}, \hat{u}, \hat{T}$ given by \eqref{eq6.1}a,b,c we state
the following result.
\begin{lemma}\label{lem:6.1}
Suppose $\rho(x,t), u(x,t)\in
\mathcal{V}_m$ and $T(x,t)\in\mathcal{W}_m$
for some $m\in\mathbb{N}$, and
\begin{subequations}\label{eq6.12}
\begin{gather}
\int_0^L(\rho_t+au_x+b\rho_x-\delta\rho_{xx}-c)v dx=0\\
\int_0^L(u_t+b_2u_x+RT_x+d\rho_x-fu_{xx}-g)v dx=0\\
\intertext {for any $v\in \mathcal{V}_m$, while }
\int_0^L(T_t+b_3T_x+hu_x-iT_{xx}-j)w dx=0
\end{gather}
\end{subequations}
for any $w\in\mathcal{W}_m$;
then, the following identity holds:
\begin{equation}\label{eq6.13}
\begin{aligned}
&\frac{1}{2}\{\|\rho(\cdot,t)\|^2_{W^{1,2}}+\|u(\cdot,t)\|^2_{W^{1,2}}
+\|T(\cdot,t)\|^2_{W^{1,2}}\}\\
&+ \int_0^t\int_0^L \Big\{i(T_{xx}^2+T_x^2)+T_t^2
+ f(u_{xx}^2+u_x^2)+u_t^2+ \delta(\rho_{xx}^2+\rho_x^2)
+\rho_t^2\Big\}\,dx\,d\tau\\
&+\frac{1}{2}\int_0^Lfu_x^2dx+\frac{1}{2}\int_0^LiT_x^2dx
\\
&=\frac{1}{2}\int_0^Lf(x,0)u_x^2(x,0)dx+\frac{1}{2}\int_0^Li(x,0)T_x^2(x,0)dx
\\
&\quad +\frac{1}{2}\{\|\rho(\cdot,0)\|^2_{W^{1,2}}+\|u(\cdot,0)\|^2_{W^{1,2}}
+\|T(\cdot,0)\|^2_{W^{1,2}}\}
\\
&\quad +\int_0^t\int_0^L \Big\{-a\rho u_x- b\rho\rho_x-b_2uu_x- RuT_x
- du\rho_x - b_3TT_x-hTu_x\\
&\quad +\frac{1}{2}f_tu_x^2+\frac{1}{2}i_tT_x^2\Big\}\,dx\,d\tau
+\int_0^t\int_0^L \Big\{au_x\rho_{xx}+b\rho_x\rho_{xx}+ b_2u_xu_{xx}\\
&\quad +RT_xu_{xx}+d\rho_xu_{xx}+b_3T_xT_{xx}
-a\rho_t u_x - b\rho_t\rho_x-b_2u_tu_x - Ru_tT_x\\
&\quad - du_t\rho_x - b_3T_tT_x-f_xu_tu_{x}-i_xT_tT_{x}
-hT_tu_x+hu_xT_{xx}\Big\}\,dx\,d\tau\\
&\quad +\int_0^t\int_0^L\Big\{ c\rho+ gu+ jT-c\rho_{xx}-gu_{xx}\\
&\quad -jT_{xx} + c\rho_t+ gu_t+ jT_t\Big\}\,dx\,d\tau
+\int_0^t\int_0^L \delta\rho_t \rho_{xx}\,dx\,d\tau
\end{aligned}
\end{equation}
\end{lemma}
\begin{remark} \label{rmk6.1} \rm
The hypotheses of Lemma \ref{lem:6.1} will hold for the Galerkin
approximations to the solution of the regularized linear initial-boundary
value problem which are constructed below.
\end{remark}
\begin{proof}
We observe that as $\rho, u\in\mathcal{V}_m$ and
$T\in\mathcal{W}_m$, all even order spatial derivatives of $\rho$, $u$ and
$T$ will be zero at $x=0$, while odd order spatial derivatives of $\rho$, $u$
and even order spatial derivatives of $T$ will vanish at $x=L$.
As $\rho\in\mathcal{V}_m$, it follows from from \eqref{eq6.12}c that
\begin{equation}\label{eq6.14}
\int_0^L\int_0^t(\rho_t+a u_x+b\rho_x-\delta\rho_{xx}-c)\rho \,dx\,d\tau=0
\end{equation}
which, after integration by parts of the term $\delta\rho\rho_{xx}$, becomes
\begin{equation}\label{eq6.15}
\begin{aligned}
&\int_0^L\frac{1}{2}\rho^2(x,t)dx-\int_0^L\frac{1}{2}\rho_0^2(x)dx
+\int_0^t\int_0^L \delta\rho_x^2\,dx\,d\tau\\
&= \int_0^t\int_0^L \{-a\rho u_x- b\rho\rho_x+ c\rho\}\,dx\,d\tau
\end{aligned}
\end{equation}
Also, as $u\in\mathcal{V}_m$ we have, as a consequence of \eqref{eq6.12}b
\begin{equation}\label{eq6.16}
\int_0^L(u_t+b_2u_x+RT_x+d\rho_x-fu_{xx}-g)udx=0
\end{equation}
If we then integrate this last result over $[0,t]$ we obtain
\begin{equation}\label{eq6.17}
\begin{aligned}
&\int_0^L\frac{1}{2}u^2(x,t)dx-\int_0^L\frac{1}{2}u_0^2(x)dx
+\int_0^t\int_0^L fu_x^2\,dx\,d\tau
\\
&= \int_0^t\int_0^L \{-b_2uu_x- RuT_x - du\rho_x+ gu \}\,dx\,d\tau
\end{aligned}
\end{equation}
Next, as $T\in\mathcal{W}_m$, it follows from \eqref{eq6.12}c that
\begin{equation}\label{eq6.18}
\int_0^L\left(T_t+b_3T_x+hu_x-iT_{xx}-j\right)Tdx=0
\end{equation}
and integrating this result over $[0,t]$ we obtain
\begin{equation}\label{eq6.19}
\begin{aligned}
&\int_0^L\frac{1}{2}T^2(x,t)dx-\int_0^L\frac{1}{2}T_0^2(x)dx
+\int_0^t\int_0^L iT_x^2\,dx\,d\tau
\\
&= \int_0^t\int_0^L\{- b_3TT_x-hTu_x+ jT\}\,dx\,d\tau\,.
\end{aligned}
\end{equation}
Since $\rho_{xx}\in \mathcal{V}_m$, \eqref{eq6.12}a yields
\begin{equation}\label{eq6.20}
\int_0^L(\rho_t+au_x+b\rho_x-c -\delta\rho_{xx})\rho_{xx}dx=0\,.
\end{equation}
Integration by parts in \eqref{eq6.20}, coupled with the conditions $\rho_x
(L, t) = 0, \rho_t (0, t) = 0$, and followed by integration over $[0, t]$,
then yields
\begin{equation}\label{eq6.21}
\begin{aligned}
&\int_0^L\frac{1}{2}\rho_x^2(x,t)dx-\int_0^L\frac{1}{2}\rho_x^2(x,0)dx
+\int_0^t\int_0^L\delta\rho_{xx}^2\,dx\,d\tau
\\
&=\int_0^t\int_0^L(au_x\rho_{xx}+b\rho_x\rho_{xx}-c\rho_{xx})\,dx\,d\tau
\end{aligned}
\end{equation}
As $u_{xx}\in \mathcal{V}_m$, \eqref{eq6.12}b implies that
\begin{equation}\label{eq6.22}
\int_0^L(u_t+b_2u_x+RT_x+d\rho_x-fu_{xx}-g)u_{xx}dx=0
\end{equation}
If we integrate by parts in this last identity, use the conditions $u_x
(L,t)=0$ and
$u_t (0,t)=0$, and then integrate over $[0,t]$, we find that
\begin{equation}\label{eq6.23}
\begin{aligned}
&\int_0^L\frac{1}{2}u_x^2(x,t)dx
-\int_0^L\frac{1}{2}u_x^2(x,0)dx+\int_0^t\int_0^Lfu^2_{xx}\,dx\,d\tau
\\
&= \int_0^t\int_0^L(b_2u_xu_{xx}+RT_xu_{xx}+d\rho_xu_{xx}-gu_{xx})\,dx\,d\tau\,.
\end{aligned}
\end{equation}
Next, as $T_{xx}\in \mathcal{W}_m$, \eqref{eq6.12}c produces
\begin{equation}\label{eq6.24}
\int_0^L(T_t+b_3T_x+hu_x-iT_{xx}-j)T_{xx}dx=0\,.
\end{equation}
Integrating by parts in \eqref{eq6.24}, using the fact that
$T_t (0,t) = T_t (L,t) = 0, t > 0$, and then integrating over $[0,t]$,
we obtain
\begin{equation}\label{eq6.25}
\begin{aligned}
&\int_0^L\frac{1}{2}T_x^2(x,t)dx-\int_0^L\frac{1}{2}T_x^2(x,0)dx
+\int_0^t\int_0^LiT^2_{xx}\,dx\,d\tau
\\
&= \int_0^t\int_0^L(b_3T_xT_{xx}+hu_xT_{xx}-jT_{xx})\,dx\,d\tau\,.
\end{aligned}
\end{equation}
Since $\rho_{t}\in \mathcal{V}_m$, \eqref{eq6.12}a yields
\begin{equation}\label{eq6.26}
\int_0^L(\rho_t+au_x+b\rho_x-c -\delta\rho_{xx})\rho_{t}dx=0\,,
\end{equation}
so that
\begin{equation}\label{eq6.27}
\int_0^t\int_0^L\rho_t^2\,dx\,d\tau
=\int_0^t\int_0^L(-a\rho_tu_x-b\rho_t\rho_x+c\rho_t
-\delta\rho_t\rho_{xx})\,dx\,d\tau
\end{equation}
Next, we note that as $u_t\in \mathcal{V}_m$, \eqref{eq6.12}b yields, after
integration over $[0,t]$,
\begin{equation}\label{eq6.28}
\int_0^t \int_0^L (u_t +b_2u_x +RT_x +d \rho_x-fu_{xx}-g) u_t dx dt=0\,.
\end{equation}
Integrating the next to the last term in \eqref{eq6.28} by parts, we find that
\begin{equation}\label{eq6.29}
\int_0^t\int_0^Lfu_{xx}u_t\,dx\,d\tau
=-\int_0^L\frac{1}{2}fu_x^2dx\Big|_0^t
+\int_0^t\int_0^L\frac{1}{2}f_tu_{x}^2\,dx\,d\tau
-\int_0^t\int_0^Lf_xu_{x}u_t\,dx\,d\tau
\end{equation}
because $u_{x}(L,t)=u_t(0,t)=0$. If we now substitute this last result back
into \eqref{eq6.29}, we obtain
\begin{equation}\label{eq6.30}
\begin{aligned}
&\int_0^t\int_0^Lu_t^2\,dx\,d\tau+\int_0^L\frac{1}{2}fu_x^2dx\\
&=\int_0^L\frac{1}{2}f(x,0)u_x^2(x,0)dx \\
&\quad +\int_0^t\int_0^L(-b_2u_tu_x-Ru_tT_x-du_t\rho_x-f_xu_tu_{x}
+\frac{1}{2}f_tu_x^2-gu_t)\,dx\,d\tau\,.
\end{aligned}
\end{equation}
Finally, as $T_t\in \mathcal{W}_m$, \eqref{eq6.12}c produces, after
integration over $[0,t]$,
\begin{equation}\label{eq6.31}
\int_0^t \int_0^L (T_t+b_3T_x+hu_x-iT_{xx}-j)T_tdx dt=0\,.
\end{equation}
Integrating the next to the last term in \eqref{eq6.31} by parts yields
\begin{equation}\label{eq6.32}
\int_0^t\int_0^LiT_{xx}T_t\,dx\,d\tau
= -\int_0^L\frac{1}{2}iT_x^2dx\Big|_0^t
+\int_0^t\int_0^L\frac{1}{2}i_tT_{x}^2\,dx\,d\tau
-\int_0^t\int_0^Li_xT_{x}u_t\,dx\,d\tau
\end{equation}
as $T_t(L,t)=T_t(0,t)=0$. Substituting this last result back into \eqref{eq6.31}
we find that
\begin{equation}\label{eq6.33}
\begin{aligned}
&\int_0^t\int_0^LT_t^2\,dx\,d\tau+\int_0^L\frac{1}{2}iT_x^2dx\\
&=\int_0^L\frac{1}{2}i(x,0)T_x^2(x,0)dx \\
&\quad +\int_0^t\int_0^L(-b_3T_tT_x-hT_tu_x-i_xT_tT_{x}
+\frac{1}{2}i_tT_x^2-jT_t)\,dx\,d\tau
\end{aligned}
\end{equation}
Adding together the results in equations \eqref{eq6.15}, \eqref{eq6.17},
\eqref{eq6.19}, \eqref{eq6.21}, \eqref{eq6.23}, \eqref{eq6.25}, \eqref{eq6.27},
\eqref{eq6.30}, and \eqref{eq6.33}, and grouping like terms together, we
obtain the result expressed by \eqref{eq6.13}.
\end{proof}
We now introduce what will turn out to be an appropriate energy functional
for the regularized system \eqref{eq6.10}a,b,c with artificial viscosity
$\delta >0$, namely, we have the following definition.
\begin{definition}\label{def:6.2} \rm
For $\rho(x,t)$, $u(x,t)$, and $T(x,t)$ we define the energy functional
\begin{equation}\label{eq6.34}
\begin{aligned}
\mathcal{E}_\delta(t)
&=\frac{1}{2}\big\{\|\rho(\cdot,t)\|^2_{W^{1,2}}+\|u(\cdot,t)\|
^2_{W^{1,2}}+\|T(\cdot,t)\|^2_{W^{1,2}}\big\}
\\
&\quad +\frac{1}{2} \int_0^t\int_0^L \Big\{i_c(T_{xx}^2+T_x^2)+T_t^2+
f_c(u_{xx}^2+u_x^2)+u_t^2 \\
&\quad + \delta(\rho_{xx}^2+\rho_x^2)+\rho_t^2 \Big\}\,dx\,d\tau
+\frac{1}{2}\int_0^Lf_cu_x^2dx+\frac{1}{2}\int_0^Li_cT_x^2dx
\end{aligned}
\end{equation}
\end{definition}
\begin{lemma}\label{lem:6.2}
Under the hypotheses of Lemma \ref{lem:6.1} we have for
$\rho,\ u\in\mathcal{V}_m$,
$T\in \mathcal{W}_m$, and $0<\delta< 1$,
\begin{equation}\label{eq6.35}
\mathcal{E}_\delta(t)
\le \mathcal{E}(0)+\mathcal{G}t+\mathcal{K}
\int_0^t \mathcal{E}_\delta(\tau)d\tau\,,
\end{equation}
where $\mathcal{G}$ and $\mathcal{K}$ are positive constants.
\end{lemma}
\begin{proof}
By Lemma \ref{lem:6.1} the identity \eqref{eq6.13} holds. The terms on the right
hand side of \eqref{eq6.13} have been separated by \{\ \} into four distinct
groups; we now proceed to estimate these terms.
In these estimates we will use generic positive constants $C_i$, $K_i$, $G_i$.
The terms from the first group involve functions and/or first derivatives with
bounded coefficients; these can be estimated as in the following sample case:
\begin{equation}\label{eq6.36}
\big| \int_0^t \int_0^L a \rho u_x \,dx\, d\tau \big|
\le K_1 \int_0^t\int_0^L
(\rho^2+u_x^2)\,dx\,d\tau, \quad K_1 = \frac{\sup|a|}{2}\,.
\end{equation}
The second group of terms involve second derivatives or time derivatives of the
functions $\rho$, $u$, and $T$. These can be estimated as follows:
for any $\eta>0$,
\begin{equation}\label{eq6.37}
\big| \int_0^t\int_0^L a\rho_{xx} u_x \,dx\,d\tau \big|
\le \eta \sup|a| \int_0^t\int_0^L\rho_{xx}^2\,dx\,d\tau
+ \frac{\sup|a| }{4\eta} \int_0^t\int_0^L u_x^2 \,dx\,d\tau\,,
\end{equation}
\[
\big| \int_0^t\int_0^L a\rho_{xx} u_x \,dx\,d\tau \big|
\le \eta C_1 \int_0^t\int_0^L\rho_{xx}^2\,dx\,d\tau+ K_2(\eta)
\int_0^t\int_0^L u_x^2 \,dx\,d\tau
\]
One further example of this type, in the first \{ \ \}, would be
\begin{equation}\label{eq6.38}
\begin{aligned}
&\big| \int_0^t\int_0^L a\rho_t u_x \,dx\,d\tau \big|
\le \eta \sup|a| \int_0^t\int_0^L\rho_t^2\,dx\,d\tau
+ \frac{\sup|a|}{4\eta} \int_0^t\int_0^L u_x^2 \,dx\,d\tau
\\
& = \eta C_2 \int_0^t\int_0^L\rho_t^2\,dx\,d\tau
+ K_4(\eta) \int_0^t\int_0^L u_x^2 \,dx\,d\tau\,.
\end{aligned}
\end{equation}
For the third group of \{ \} terms on the right-hand side of \eqref{eq6.13}
we have estimates which conform to the pattern in the following two
examples:
\begin{gather}\label{eq6.39}
\begin{aligned}
\big| \int_0^t\int_0^L c\rho \,dx\,d\tau \big|
&\le \frac{1}{2}\int_0^t\int_0^L\rho^2\,dx\,d\tau
+ \frac{1}{2} \int_0^t\int_0^Lc^2\,dx\,d\tau\\
&\le K_3\int_0^t\int_0^L\rho^2\,dx\,d\tau+G_1\cdot t\,,
\end{aligned}\\
\label{eq6.40}
\begin{aligned}
\big| \int_0^t\int_0^L c\rho_{xx} \,dx\,d\tau \big|
&\le \eta\int_0^t\int_0^L\rho_{xx}^2\,dx\,d\tau
+ \frac{1}{4\eta} \int_0^t\int_0^Lc^2\,dx\,d\tau \\
&\le \eta\int_0^t\int_0^L\rho_{xx}^2\,dx\,d\tau+ G_2 (\eta)\cdot t, \quad
G_2 (\eta) = \frac{{\rm const.}}{\eta}
\end{aligned}
\end{gather}
Terms involving second order spatial derivatives or time derivatives
on the right hand side of, say, \eqref{eq6.40} can be moved to the
left hand side of \eqref{eq6.13} to be absorbed by those terms with a
similar structure, if $\eta$ is chosen sufficiently small.
Finally the last term on the right hand side of \eqref{eq6.13} may be
estimated as follows:
\begin{equation}\label{eq6.41}
\big| \int_0^t\int_0^L \delta\rho_t\rho_{xx} \,dx\,d\tau \big|
\le \int_0^t\int_0^L \frac{\delta}{2}\rho_t^2\,dx\,d\tau
+\int_0^t\int_0^L \frac{\delta}{2}\rho^2_{xx} \,dx\,d\tau\,.
\end{equation}
Both terms on the right-hand side of \eqref{eq6.41} can be brought over
to the left-hand side of
\eqref{eq6.13} and absorbed by those terms with a similar structure; this is
true for the first term on the right-hand side of \eqref{eq6.41} provided
$\delta <2$.
We observe that on the left-hand side of \eqref{eq6.13},
\begin{subequations}\label{eq6.42}
\begin{gather}
\int_0^t\int_0^LiT_{xx}^2\,dx\,d\tau
\ge\int_0^t\int_0^Li_cT_{xx}^2\,dx\,d\tau \\
\intertext{and similarly for term involving $fu_{xx}^2$. Then, for $\eta$ chosen small enough}
\int_0^t\int_0^L(i_c-C\eta)T_{xx}^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{2}i_cT_{xx}^2\,dx\,d\tau \\
\int_0^t\int_0^L(f_c-C\eta)u_{xx}^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{2}f_cu_{xx}^2\,dx\,d\tau \\
\int_0^t\int_0^L\left(\frac{\delta}{2}-C\eta\right)\rho_{xx}^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{4}\delta\rho_{xx}^2\,dx\,d\tau \\
\int_0^t\int_0^L(1-C\eta)T_t^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{2}T_t^2\,dx\,d\tau
\\
\int_0^t\int_0^L(1-C\eta)u_t^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{2}u_t^2\,dx\,d\tau
\\
\int_0^t\int_0^L\left(\frac{1}{2}-C\eta\right)\rho_t^2\,dx\,d\tau
\ge \int_0^t\int_0^L\frac{1}{4}\rho_t^2\,dx\,d\tau\,,
\end{gather}
\end{subequations}
where $C=\sum C_i$.
We note that once $\eta$ is chosen, the $K_i$ and $G_i$, $i=1,2,\dots$
are constants.
Adding all our estimates, and making use of \eqref{eq6.42}a-g, we obtain
\eqref{eq6.35} with $\mathcal{K} = \sum K_i$ and $\mathcal{G}=\sum G_i$.
\end{proof}
As a consequence of Lemma \ref{lem:6.2}, we have the following a priori
estimate for the energy functional $\mathcal{E}_{\delta} (t)$.
\begin{lemma}\label{lem:6.3}
For some $C_{t_0}>0$, and all $t$, $0\le t\le t_0$, we have,
under the hypotheses
of Lemma \ref{lem:6.2},
\begin{equation}\label{eq6.43}
\mathcal{E}_\delta(t)\le C_{t_0}
\Big(\frac{\mathcal{G}}{\mathcal{K}}+\mathcal{E}(0)\Big)
\end{equation}
\end{lemma}
\begin{proof}
The proof is a consequence of Gronwall's inequality \cite{e1}.
By virtue of \eqref{eq6.35},
\begin{equation}\label{eq6.44}
\mathcal{E}_\delta(t)+\frac{\mathcal{G}}{\mathcal{K}}
\le\mathcal{E}(0)+\frac{\mathcal{G}}{\mathcal{K}}+\mathcal{G}t
+\mathcal{K}\int_0^t\mathcal{E}_\delta(\tau)d\tau, \quad 0\le t\le t_0\,.
\end{equation}
If we set
\begin{equation}\label{eq6.45}
\bar{\mathcal{E}}_\delta(t)
=\mathcal{E}_\delta(t)+\frac{\mathcal{G}}{\mathcal{K}}
\end{equation}
it follows that
\begin{equation}\label{eq6.46}
\bar{\mathcal{E}}_\delta(t)\le\bar{\mathcal{E}}(0)
+\mathcal{K}\int_0^t\bar{\mathcal{E}}_\delta(\tau)d\tau
\end{equation}
Applying Gronwall's inequality to $\bar{\mathcal{E}}_\delta$ we obtain
\begin{equation}\label{eq6.47}
\mathcal{E}_\delta(t)\le \mathcal{E}_\delta(t)
+\frac{\mathcal{G}}{\mathcal{K}} \le
\Big(\frac{\mathcal{G}}{\mathcal{K}}+\mathcal{E}(0)\Big)e^{\mathcal{K}t}, \quad
0\le t\le t_0
\end{equation}
from which \eqref{eq6.43} follows, for $0 \leq t \leq t_0$, with
$C_{t_0} =\exp (\mathcal{K} t_0)$.
\end{proof}
As a prelude to the introduction of the Galerkin approximations, we first extend
the initial data symmetrically to $[0, 2L]$; i.e., for $L\le x\le 2L$
we define
\begin{subequations}\label{eq6.48}
\begin{gather}
u_0(x)=u_0(2L-x) \\
\rho_0(x)=\rho_0(2L-x)\\
T_0(x)=T_0(2L-x)
\end{gather}
\end{subequations}
and then extend $u_0$, $\rho_0$, $T_0$ periodically to the entire line with
period $2L$. A complete orthonormal set of functions on $[0, 2L]$, with
respect to the inner product $ = \int_0^{2L} fg dx$, is given by
\begin{equation}\label{eq6.49}
\big\{\frac{1}{\sqrt{2L}},\, \frac{1}{\sqrt{L}}\sin\frac{l\pi x}{2L},\,
\frac{1}{\sqrt{L}}\cos\frac{l\pi x}{2L};\, l=1,2,\dots\}
\end{equation}
We also observe that the set
\begin{equation}\label{eq6.50}
\{\sqrt\frac{2}{L}\sin\frac {(2l+1)\pi x}{2L},\; l=0,1,2,\dots\}
\end{equation}
is a complete orthonormal set with respect to
$$
\{u(\cdot)|u(\cdot)\in L^2[0,L],\ u(0)=0,\ u_x(L)=0\}.
$$
in the trace sense \cite{a1,e1}. Similarly,
\begin{equation}\label{eq6.51}
\{\sqrt\frac{2}{L}\sin\frac {l\pi x}{L},\; l=0,1,2,\dots\}
\end{equation}
is a complete orthonormal set with respect to
$$
\{u(\cdot)|u(\cdot)\in L^2[0,L],\ u(0)=0,\ u(L)=0\}.
$$
in the trace sense.
To proceed, we define candidate ``approximate solutions"; i.e., Galerkin
approximations for the system consisting of \eqref{eq6.10}a,b,c,
\eqref{eq6.7}a,b,c and \eqref{eq6.8}a-f of the form
\begin{subequations}\label{eq6.52}
\begin{gather}
\rho_{m}=\sum_{l=0}^m A_{lm}(t)\sqrt\frac{2}{L}\sin\frac {(2l+1)\pi x}{2L}\\
u_{m}=\sum_{l=0}^m B_{lm}(t)\sqrt\frac{2}{L}\sin\frac {(2l+1)\pi x}{2L}\\
T_{m}=\sum_{l=0}^m C_{lm}(t)\sqrt\frac{2}{L}\sin\frac {l\pi x}{L}\,,
\end{gather}
\end{subequations}
where for $0 \leq l \leq m, m = 0, 1, 2, \dots$,
\begin{subequations}\label{eq6.53}
\begin{gather}
A_{lm}(0)=\xi_l\\
B_{lm}(0)=\eta_l\\
C_{lm}(0)=\nu_l
\end{gather}
\end{subequations}
and $\xi_l$, $\eta_l$, $\nu_l$ are determined as the coefficients in the
following expansions of the initial data:
\begin{subequations}\label{eq6.54}
\begin{gather}
\sum_{l=0}^m\xi_l\sqrt\frac{2}{L}\sin\frac {(2l+1)\pi x}{2L}\to \rho_0(x),
\quad\text{in } L^2(0,L)\\
\sum_{l=0}^m\eta_l\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}\to u_0(x),
\quad\text{in } L^2(0,L)\\
\sum_{l=0}^m\nu_l\sqrt\frac{2}{L} \sin\frac {l\pi x}{L}\to T_0(x),
\quad\text{in } L^2(0,L),
\end{gather}
\end{subequations}
as $m\to \infty$. The $\xi_l$, $\eta_l$ and $\nu_l$ are uniquely
determined,
once the functions $\rho_0(x)$, $u_0(x)$, and $T_0(x)$ have been extended as
described above. In \eqref{eq6.54}a,b,c, $\rho_0(x)$, $u_0(x)$, and $T_0(x)$
are actually the functions $\hat{\rho}_0(x)$, $\hat{u}_0(x)$, $\hat{T}_0(x)$, the
hats having been dropped.
We now require that the coefficients $A_{km}(t)$, $B_{km}(t)$, and $C_{km}(t)$
satisfy the linear system of ordinary differential equations:
\begin{subequations}\label{eq6.55}
\begin{gather}
\begin{aligned}
A_{km}'(t) &=-\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L^2}
\langle a(x,t)\cos\frac {(2l+1)\pi x}{2L}, \sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad -\sum_{l=0}^m A_{lm}(t)\frac{(2l+1)\pi}{2L^2}\langle b(x,t)\cos\frac {(2l+1)\pi x}{2L}, \sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad - A_{km}(t)\delta \frac{(2l+1)^2\pi^2}{2L^3} +\langle c(x,t),
\frac{1}{\sqrt{L}} \sin\frac {(2k+1)\pi x}{2L}\rangle \,,
\end{aligned} \\
\begin{aligned}
& B_{km}'(t)\\
&=-\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L^2}
\langle b_2(x,t)\cos\frac {(2l+1)\pi x}{2L},\sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad -\sum_{l=0}^m A_{lm}(t)\frac{(2l+1)\pi}{2L^2}\langle d(x,t)
\cos\frac {(2l+1)\pi x}{2L}, \sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad -\sum_{l=0}^m C_{lm}(t)R\frac{l\pi}{L^2}\langle
\cos\frac {l\pi x}{L}, \sin\frac {(2k+1)\pi x}{2L}\rangle
+\langle g(x,t), \frac{1}{\sqrt{L}}\sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad -\sum_{l=0}^m B_{lm}(t)\frac {(2l+1)^2\pi^2 }{4L^3}\langle f(x,t)
\sin\frac {(2l+1)\pi x}{2L}, \sin\frac {(2k+1)\pi x}{2L}\rangle \,,
\end{aligned}\\
\begin{aligned}
C_{km}'(t)&=-\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L^2}
\langle h(x,t)\cos\frac {(2l+1)\pi x}{2L},\sin\frac {k\pi x}{L}\rangle \\
&\quad -\sum_{l=0}^m C_{lm}(t)\frac{l\pi}{L^2}\langle b_3(x,t)
\cos\frac {l\pi x}{L},\sin\frac {k\pi x}{L}\rangle \\
&\quad -\sum_{l=0}^m C_{lm}(t)\frac {l^2\pi^2 }{L^3}\langle i(x,t)
\sin\frac {l\pi x}{2L},\sin\frac {k\pi x}{L}\rangle \\
&\quad +\langle j(x,t),\frac{1}{\sqrt{L}}\sin\frac {k\pi x}{L}\rangle
\end{aligned}
\end{gather}
\end{subequations}
and the initial conditions, for $ 0\le l\le m$,
\begin{subequations}\label{eq6.56}
\begin{gather}
A_{lm}(0)=\xi_l\\
B_{lm}(0)=\eta_l\\
C_{lm}(0)=\nu_l
\end{gather}
\end{subequations}
By standard ODE theory the system \eqref{eq6.55}a,b,c, \eqref{eq6.56}a,b,c
possesses a unique solution. Moreover, we have the following result.
\begin{lemma}\label{lem:6.4}
Let $\{A_{km}(t), B_{km}(t), C_{km}(t)\}$, $0\le k \le m$, be the unique
solution of the initial value problem \eqref{eq6.55}a,b,c,
\eqref{eq6.56}a,b,c. Then the ``approximate solutions" $\rho_m$, $u_m$ and $T_m$,
as defined by \eqref{eq6.52}a,b,c, satisfy the hypotheses of Lemma \ref{lem:6.1}
(and, hence, those of Lemma \ref{lem:6.2} as well).
\end{lemma}
\begin{proof}
We have
\begin{equation}\label{eq6.57}
\begin{aligned}
&\rho_{mt}+au_{mx}+b\rho_{mx}-\delta\rho_{mxx}-c\\
&=\sum_{l=0}^m A_{lm}'(t)\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}
+\sum_{l=0}^m B_{lm}(t)a(x,t)\frac{(2l+1)\pi}{2L}\sqrt\frac{2}{L}
\cos\frac {(2l+1)\pi x}{2L} \\
&\quad +\sum_{l=0}^m A_{lm}(t)b(x,t)\frac{(2l+1)\pi}{2L}\sqrt\frac{2}{L}
\cos\frac {(2l+1)\pi x}{2L}\\
&\quad -c(x,t)+\sum_{l=0}^m A_{lm}(t)\delta\frac{(2l+1)^2\pi^2}{4L^2}
\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}
\end{aligned}
\end{equation}
If we now multiply \eqref{eq6.57} by
$\sqrt\frac{2}{L} \sin\frac{(2k+1)\pi x}{2L}$,
$0\le k \le m $, and integrate from 0 to $L$, we obtain, for $0 \leq k \leq m$,
\begin{equation}\label{eq6.58}
\begin{aligned}
&\int_0^L(\rho_{mt}+au_{mx}+b\rho_{mx}-\delta\rho_{mxx}-c)
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}dx\\
&= A_{km}'(t) +\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L}
\langle a(x,t)\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L},
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad +\sum_{l=0}^m A_{lm}(t)\frac{(2l+1)\pi}{2L}\langle b(x,t)
\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L},
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle \\
&\quad +A_{km}(t)\delta\frac{(2l+1)^2\pi^2}{2L^3}
-\langle c(x,t),\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle =0
\end{aligned}
\end{equation}
(for each $0\le k \le m$) as a consequence \eqref{eq6.55}a. Therefore, for
any $v \in \mathcal{V}_m$,
\begin{equation}\label{eq6.59}
\int_0^L(\rho_{mt}+au_{mx}+b\rho_{mx}-\delta\rho_{mxx}-c)vdx=0\,.
\end{equation}
Next, we compute that
\begin{equation}\label{eq6.60}
\begin{aligned}
&u_{mt}+b_2u_{mx}+RT_{mx}+d\rho_{mx}-fu_{mxx}-g\\
&=\sum_{l=0}^m B_{lm}'(t)\sqrt\frac{2}{L}
\sin\frac {(2l+1)\pi x}{2L}+\sum_{l=0}^m B_{lm}(t)b_2(x,t)
\frac{(2l+1)\pi}{2L}\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L} \\
&\quad +\sum_{l=0}^m C_{lm}(t)R\frac{l\pi}{L}\sqrt\frac{2}{L}
\cos\frac {l\pi x}{L}+\sum_{l=0}^m A_{lm}(t)d(x,t)
\frac{(2l+1)\pi}{2L}\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L}\\
&\quad +\sum_{l=0}^m B_{lm}(t)\left(\frac {(2l+1)\pi }{2L}\right)^2f(x,t)
\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}-g(x,t)
\end{aligned}
\end{equation}
Multiplying \eqref{eq6.60} by $\sqrt\frac{2}{L} \sin\frac{(2k+1)\pi x}{2L}$,
$0\le k\le m$, and integrating over $[0,L]$, we obtain, for $0 \leq k \leq
m$,
\begin{align}
&\int_0^L(u_{mt}+b_2u_{mx}+RT_{mx}+d\rho_{mx}-fu_{mxx}-g)\sqrt\frac{2}{L}
\sin\frac {(2k+1)\pi x}{2L}dx \nonumber \\
&= \sum_{l=0}^m B_{km}'(t) \nonumber \\
&\quad +\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L}
\langle b_2(x,t)\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L},
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle \nonumber \\
&\quad +\sum_{l=0}^m A_{lm}(t)\frac{(2l+1)\pi}{2L}\langle d(x,t)
\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L}, \sqrt\frac{2}{L}
\sin\frac {(2k+1)\pi x}{2L}\rangle \nonumber \\
&\quad +\sum_{l=0}^m C_{lm}(t)R\frac{l\pi}{L}\langle \sqrt\frac{2}{L}
\cos\frac {l\pi x}{L}, \sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle
\nonumber \\
&\quad -\langle g(x,t), \sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle
\nonumber \\
&\quad +\sum_{l=0}^m B_{lm}(t)\left(\frac {(2l+1)\pi }{2L}\right)^2
\langle f(x,t)\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L},
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}\rangle \nonumber \\
& =0 \label{eq6.61}
\end{align}
as a consequence of \eqref{eq6.55}b. Thus, for any $v \in \mathcal{V}_m$,
\begin{equation}\label{eq6.62}
\int_0^L(u_{mt}+b_2u_{mx}+RT_{mx}+d\rho_{mx}-fu_{mxx}-g)
\sqrt\frac{2}{L} \sin\frac {(2k+1)\pi x}{2L}dx=0\,.
\end{equation}
Finally, we have
\begin{equation}\label{eq6.63}
\begin{aligned}
& T_t+b_3T_x+hu_x-iT_{xx}-j \\
& =\sum_{l=0}^m C_{lm}'(t)\sqrt\frac{2}{L} \sin\frac {l\pi x}{L}
+\sum_{l=0}^m B_{lm}(t)h(x,t)\frac{(2l+1)\pi}{2L}\sqrt\frac{2}{L}
\cos\frac {(2l+1)\pi x}{2L} \\
&\quad +\sum_{l=0}^m C_{lm}(t)\frac{l\pi}{L}b_3(x,t)\sqrt\frac{2}{L}
\cos\frac{l\pi x}{L}\\
&\quad +\sum_{l=0}^m C_{lm}(t)\left(\frac{l\pi }{L}\right)^2i(x,t)\sqrt\frac{2}{L}
\sin\frac {l\pi x}{2L}-j(x,t)
\end{aligned}
\end{equation}
Multiplying \eqref{eq6.63} by $\sqrt\frac{2}{L} \sin\frac {k\pi x}{L}$,
$0\le k \le m$, and integrating over $[0,L]$, we obtain
\begin{equation}\label{eq6.64}
\begin{aligned}
&\int_0^L(T_t+b_3T_x+hu_x-iT_{xx}-j)\sqrt\frac{2}{L} \sin\frac {k\pi x}{L}dx\\
&=\sum_{l=0}^m C_{km}'(t)\\
&\quad +\sum_{l=0}^m B_{lm}(t)\frac{(2l+1)\pi}{2L}
\langle h(x,t)\sqrt\frac{2}{L} \cos\frac {(2l+1)\pi x}{2L},\sqrt\frac{2}{L}
\sin\frac {k\pi x}{L}\rangle \\
&\quad +\sum_{l=0}^m C_{lm}(t)\frac{l\pi}{L}\langle b_3(x,t)
\sqrt\frac{2}{L} \cos\frac {l\pi x}{L},\sqrt\frac{2}{L}
\sin\frac {k\pi x}{L}\rangle \\
&\quad +\sum_{l=0}^m C_{lm}(t)\left(\frac {l\pi }{L}\right)^2
\langle i(x,t)\sqrt\frac{2}{L} \sin\frac {l\pi x}{2L},\sqrt\frac{2}{L}
\sin\frac {k\pi x}{L}\rangle\\
&\quad -\langle j(x,t),\sqrt\frac{2}{L} \sin\frac {k\pi x}{L}\rangle =0\,,
\end{aligned}
\end{equation}
as a consequence of \eqref{eq6.55}c. Thus, we conclude that for any $w \in
\mathcal{W}_m$,
\begin{equation}\label{eq6.65}
\int_0^L(T_t+b_3T_x+hu_x-iT_{xx}-j)wdx=0
\end{equation}
which completes the proof of Lemma \ref{lem:6.4}
\end{proof}
We are now in a position to state the main result of this section.
\begin{theorem}\label{theor:6.1}
The initial-boundary value problem \eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c,
\eqref{eq6.8}a-f, has a unique solution $(\rho, u, T)$, for any $t >0$, such
that
\begin{subequations}\label{eq6.66}
\begin{gather}
\rho, u, T \in L^2 ([0,t]; \ W^{2,2} [0,L]),\\
\rho_t, u_t, T_t \in L^2 ([0,t]; \ L^2 [0,L])\,.
\end{gather}
\end{subequations}
\end{theorem}
\begin{proof}
The approximations $\rho_m$, $u_m$, and $T_m$ defined by \eqref{eq6.52}a,b,c
satisfy the hypotheses of Lemma \ref{lem:6.3}. Therefore, the {\it a priori}
estimate \eqref{eq6.43} applies to $\rho_m$, $u_m$, and $T_m$; i.e., on any
interval $[0,t_0]$ we have
\begin{equation}\label{eq6.67}
\mathcal{E}_{\delta m}(t)\le C_{t_0}
\Big(\frac{\mathcal{G}}{\mathcal{K}}+\mathcal{E}_{m}(0)\Big)
\end{equation}
Therefore,
\begin{subequations}\label{eq6.68}
\begin{gather}
\rho_{m}, \ u_{m},\ T_{m}\ \in L^2([0,t];W^{2,2}[0,L]);\\
\rho_{m}, \ u_{m},\ T_{m}\ \in W^{1,2}([0,t];L^2[0,L])
\end{gather}
\end{subequations}
for $0\le t\le t_0$, which implies that $\rho_{m}$, $u_{m}$, and $T_{m}$ are
continuous in both time and space. By the choice at $t=0$ of the coefficients
in the Galerkin approximations; i.e., \eqref{eq6.53}a,b,c, $\rho_{m}$, $u_{m}$,
and $T_{m}$ converge as $t\to 0$ to $\rho_{0}(x)$, $u_{0}(x)$, and
$T_{0}(x)$, so by the continuity of $\rho_{m}$, $u_{m}$, and $T_{m}$ with
respect to time we conclude that $\mathcal{E}_{m}(0)\to \mathcal{E}(0)$.
Thus, $\mathcal{E}_{m}(0)$ is a bounded sequence and \eqref{eq6.67} then implies
that $\mathcal{E}_{\delta m}(t)$ is also a bounded sequence, for $0\le t\le t_0$.
Therefore, as each of the sequences $\rho_{m}$, $u_{m}$, and $T_{m}$ is bounded
in $L^2([0,t];W^{2,2}[0,L])$ they have convergent subsequences $\rho_{m_k}$,
$u_{m_k}$, and $T_{m_k}$ (which we will also denote as $\rho_{m}$, $u_{m}$,
and $T_{m}$) that converge weakly in $L^2([0,t];W^{2,2}[0,L])$ to unique limits
$\rho$, $u$, and $T$
We want to show that the limiting set $\{ \rho, u,T \}$ is a solution of the
initial-boundary value problem \eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c,
\eqref{eq6.8}a-f, with \eqref{eq6.10}a,b,c being satisfied in the sense of
distributions.
We observe that $\{\rho_{m}, u_{m}, T_{m}\}$ satisfy
\begin{subequations}\label{eq6.69}
\begin{gather}
\int_0^t\int_0^L(\rho_{mt}+au_{mx}+b\rho_{mx}-\delta\rho_{mxx}-c)v \,dx\,d\tau=0,\\
\int_0^t\int_0^L(u_{mt}+b_2u_{mx}+RT_{mx}+d\rho_{mx}-fu_{mxx}-g)v \,dx\,d\tau=0
\end{gather}
for any $v\in \mathcal{V}_m$ with coefficients $\alpha_l(t)\in C^1(0,t_0)$;
while
\begin{equation}
\int_0^t\int_0^L(T_{mt}+b_3T_{mx}+hu_{mx}-iT_{mxx}-j)w \,dx\,d\tau=0
\end{equation}
\end{subequations}
for any $w\in \mathcal{W}_m$, with coefficients $\beta(t)\in C^1(0,t_0)$, for
$m=1,2,\ldots$.
Consider an arbitrary function $\psi \in L^2([0,t_0]; C_0^\infty[0,L])$ with
compact support on $[0,t_0]\times[0,L]$; this function can be expanded
in a series
\begin{equation}\label{eq6.70}
\psi(x,t)=\sum_{l=0}^\infty p_l(t)\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}
\end{equation}
which converges uniformly to $\psi$. The series obtained by differentiating
\eqref{eq6.70} term by term $k$ times also converges uniformly to the respective
$k^{th}$ derivative of $\psi$.
For the approximation to $\psi$ given by
\begin{equation}\label{eq6.71}
\psi_n(x,t)=\sum_{l=0}^n p_l(t)\sqrt\frac{2}{L} \sin\frac {(2l+1)\pi x}{2L}\,,
\end{equation}
we have $\psi_n\to \psi$ in $L^2([0,t_0]; C_0^\infty[0,L])$, as
$n\to\infty$. We observe that $\psi_n\in \mathcal{V}_m$, for $m\ge n$,
with $p_l(t)=0$, for $n0$,
whose existence was established in Theorem \ref{theor:6.1}. Then, in fact,
\begin{equation}\label{eq6.81}
\rho, u, T \in C^\infty([0,t]; C^\infty[0,L])
\end{equation}
\end{theorem}
\begin{proof}
We rewrite system \eqref{eq6.10}a,b,c in the form
\begin{subequations}\label{eq6.82}
\begin{gather}
\rho_t-\delta\rho_{xx}+au_x+b\rho_x=c\\
u_t-fu_{xx}+b_2u_x+RT_x+d\rho_x=g\\
T_t-iT_{xx}+b_3T_x+hu_x=j
\end{gather}
\end{subequations}
As $c,g,j\in L^2([0,t]; W^{1,2}[0,L])$, standard parabolic
theory \cite{l1} impplies that $\rho, u, T$ belong to $L^2([0,t]; W^{3,2}[0,L])$
and, because
$\rho, u, T\in L^2([0,t]; W^{3,2}[0,L])$, we can differentiate the equations in
\eqref{eq6.10}a,b,c with respect to $x$ to obtain the system
\begin{subequations}\label{eq6.83}
\begin{gather}
(\rho_x)_t-\delta(\rho_x)_{xx}+a(u_x)_x+b(\rho_x)_x=-a_xu_x-b_x\rho_x+c_x\\
(u_x)_t-f(u_x)_{xx}+b_2(u_x)_x+R(T_x)_x+d(\rho_x)_x
=-f_xu_{xx}-b_{2x}u_x-d_x\rho_x+g_x\\
(T_x)_t-i(T_x)_{xx}+b_3(T_x)_x+h(u_x)_x=-i_xT_{xx}-b_{3x}T_x-h_xu_x+j_x
\end{gather}
\end{subequations}
which is a system of equations for $\rho_x,u_x, T_x$ with the same principal
part as \eqref{eq6.82}a,b,c. Also, each forcing term on the right hand side of
\eqref{eq6.83}a,b,c is again in $L^2([0,t]; W^{1,2}[0,L])$. Therefore,
$\rho_x, u_x, T_x \in L^2([0,t]; W^{3,2}[0,L])$, in which
case $\rho, u, T \in L^2([0,t]; W^{4,2}[0,L])$.
By differentiating \eqref{eq6.82}a,b,c with respect to time,
we obtain a system of
parabolic equations for $\rho_t, u_t$ and $T_t$ which also has the same
principal part as \eqref{eq6.82}a,b,c and for this system each forcing term is,
again, in $ L^2([0,t]; W^{1,2}[0,L])$. This leads to the result that
$\rho_t, u_t, T_t \in L^2([0,t]; W^{3,2}[0,L])$. By continuing this
argument we may establish that the spatial, time, and mixed derivatives
of $\rho$, $u$, and $T$, of all orders, are in $L^2([0,t]; W^{3,2}[0,L])$
which, in turn, implies the result \eqref{eq6.81}.
\end{proof}
\section{Existence of solutions to the Linear hyperbolic-parabolic
initial-boundary value problem}
In this section we will establish
existence of a unique solution for the mixed hyperbolic-parabolic initial
boundary value problem \eqref{eq6.6}a,b,c, \eqref{eq6.7}a,b,c,
\eqref{eq6.8}a-e. Our assumptions on the coefficients in \eqref{eq6.6}a,b,c
are those stated as \eqref{eq6.9}a,b,c; in addition, we will require
that
\begin{subequations}\label{eq7.1}
\begin{gather}
c(0,t)=c(L,t) = 0,t > 0, \\
b(0,t)\geq \zeta, \quad \text{for some $\zeta >0$ and all $t>0$.}
\end{gather}
\end{subequations}
For the coefficient $c(x,t)$ defined by the nonlinear problem; i.e.,
\eqref{eq6.3}e, \eqref{eq7.1}a is satisfied as $(\ln A)_x =0$ at $x=0,L,
\forall t \geq 0$; however, \eqref{eq7.1}b is not satisfied for $b(x,t)$ as
defined by \eqref{eq6.3}b. This is, however, of little concern here as the
results presented in \cite{b3}, for the original nonlinear problem, depend only on
the existence and uniqueness theorem proven in \S6 for the linear system
with artificial viscosity without the hypotheses \eqref{eq7.1}a,b. In the
present section, we will again prove an existence and uniqueness theorem for
the problem \eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c, \eqref{eq6.8}a-f;
however,
the addition of the hypotheses \eqref{eq7.1}a,b will enable us to use an
energy functional which does not depend explicitly on the artificial
viscosity parameter $\delta$ and this, in turn, will allow us to extract the
limit, as $\delta \to 0$, of the solutions of the regularized
problem so as to obtain the desired solution of \eqref{eq6.6}a,b,c,
\eqref{eq6.7}a,b,c, \eqref{eq6.8}a-e.
We begin with the following definition.
\begin{definition}\label{def:7.1} \rm
For $\rho(x,t), u(x,t)$, and $T(x,t)$ we define the energy functional
\begin{equation}\label{eq7.2}
\begin{aligned}
\hat{\mathcal{E}_{\delta}} (t)
&=\frac{1}{2} \{ \| \rho(\cdot, t)
\|^{2}_{W^{1,2}} + \| u(\cdot,t) \|^{2}_{W^{1,2}} + \| T(\cdot,t)
\|^{2}_{W^{1,2}} \} \\
&\quad +\frac{1}{2} \int_0^t \int_0^L \{ i_c (T^2_{xx} + T^2_x) + T_t^2 +
f_c (u^2_{xx} + u^2_x) + u^2_t + \rho^2_t \} \,dx\,d\tau
\\
&\quad +\frac{1}{2} \int_0^L f_c u^2_x dx + \frac{1}{2} \int_0^L i_c T^2_x dx
\end{aligned}
\end{equation}
\end{definition}
We note that $\hat{\mathcal{E}_{\delta}} (t)$ is almost identical with
$\mathcal{E}_{\delta} (t)$ in \eqref{eq6.34}, except that in
$\hat{\mathcal{E}_{\delta}} (t)$ the term
$$
\frac{\delta}{2} \int_0^t \int_0^L (\rho_x^2 + \rho^2_{xx}) \,dx\,d\tau
$$
has been deleted; thus $\hat{\mathcal{E}_{\delta}} (t)$ depends, implicitly,
on $\delta$ because $\rho, u, T$ eventually will (as solutions of the
regularized linear problem) but $\hat{\mathcal{E}_{\delta}}$ does not depend explicitly on $\delta$. With
$\mathcal{V}_m$ and $\mathcal{W}_m$ defined as in \S6; i.e.,
\eqref{eq6.11}a,b we now have the following counterpart to Lemma
\ref{lem:6.1}:
\begin{lemma}\label{lem:7.1}
Suppose $\rho(x,t), u(x,t) \in \mathcal{V}_m$ and $T(x,t) \in \mathcal{W}_m$
for some $m \in \mathbb{N}$, and
\begin{subequations}\label{eq7.3}
\begin{gather}
\int_0^L (\rho_t + a u_x + b \rho_x - \delta \rho_{xx} - c) v dx = 0 \\
\int_0^L (u_t + b_2 u_x + RT_x + d\rho_x - f u_{xx} - g) vdx = 0 \\
\intertext{for any $v \in \mathcal{V}_m$, while}
\int_0^L (T_t + b_3 T_x + h u_x - i T_{xx} - j) wdx = 0
\end{gather}
\end{subequations}
for any $w \in \mathcal{W}_m$; then the following identity holds:
\begin{align}
&\frac{1}{2} \{ \| \rho(\cdot,t) \|^{2}_{W^{1,2}} + \| u(\cdot,t)
\|^{2}_{W^{1,2}} + \| T(\cdot,t) \|^2_{W^{1,2}} \} \nonumber \\
& + \int_0^t \int_0^L \{ i(T^2_{xx} + T_x^2) + T_t^2 + f(u^2_{xx} +
u_x^2) + u_t^2 + \delta (\rho^2_{xx} + \rho_x^2) + \rho_t^2 \} \,dx\,d\tau
\nonumber \\
&+ \frac{1}{2} \int_0^L f u_x^2 dx + \frac{1}{2} \int_0^L i T_x^2 dx +
\frac{1}{2} \int_0^t b(0, \tau) \rho_x (0, \tau)^2 d\tau \nonumber \\
&= -\int_0^t a(0, \tau) u_x (0, \tau) \rho_x (0, \tau) d\tau + \frac{1}{2}
\int_0^L f(x,0) u_x^2 (x,0) dx \nonumber \\
&\quad + \frac{1}{2} \int_0^L i(x,0) T_x^2 (x,0)dx
+ \frac{1}{2} \{ \| \rho(\cdot,0) \|^2_{W^{1,2}} + \| u(\cdot,0)
\|^2_{W^{1,2}} + \| T(\cdot,0) \|^2_{W^{1,2}} \}
\nonumber \\
&\quad + \int_0^t \int_0^L \Big\{ -a \rho u_x -b \rho \rho_x - b_2 u u_x - RuT_x
- du \rho_x - b_3 TT_x - hT u_x \nonumber \\
&\quad + \frac{1}{2} f_t u^2_x + \frac{1}{2} i_t
T_x^2 - \frac{1}{2} b_x \rho_x^2 \Big\} dx\, d\tau
+ \int_0^t \int_0^L \Big\{ -au_{xx} \rho_x + b_2 u_x u_{xx} \nonumber\\
&\quad + RT_x u_{xx}+ d \rho_x u_{xx}
+ b_3 T_x T_{xx} - a \rho_t u_x
-b \rho_t \rho_x -b_2 u_t u_x - Ru_t T_x \nonumber\\
&\quad - du_t \rho_x - b_3 T_t T_x - f_x u_t u_x
- i_x T_t T_x - h T_t u_x + hu_x T_{xx} \Big\} \,dx\,d\tau \nonumber \\
&\quad + \int_0^t \int_0^L \{ c\rho + gu + jT - c \rho_{xx} - gu_{xx} -
jT_{xx} + c \rho_t + gu_t + jT_t \} \,dx\,d\tau \nonumber\\
&\quad + \int_0^t \int_0^L
\delta \rho_t \rho_{xx} \,dx\,d\tau\,. \label{eq7.4}
\end{align}
\end{lemma}
\begin{proof}
We observe that as $\rho, u \in \mathcal{V}_m$ and $T \in
\mathcal{W}_m$, all even order spatial derivatives of $\rho, u$ and $T$ will
be zero at $x=0$, while odd order spatial derivatives of $\rho, u$ and even
order spatial derivatives of $T$ will vanish at $x=L$.
As $\rho \in \mathcal{V}_m$, it follows from \eqref{eq7.3}a that
$$
\int_0^L \int_0^t (\rho_t + au_x + b \rho_x - \delta \rho_{xx} - c)
\rho \,dx\,d \tau = 0
$$
or
$$
\int_0^L \frac{1}{2} \rho^2 (t,x)dx - \int_0^L \frac{1}{2} \rho_0^2
(x)dx = \int_0^t \int_0^L \{ -a \rho u_x -b \rho \rho_x + c \rho + \delta
\rho_{xx} \rho \} \,dx\,d\tau
$$
which, after integration by parts of the term $\delta \rho \rho_{xx}$,
becomes
\begin{equation}\label{eq7.5}
\begin{aligned}
&\int_0^L \frac{1}{2} \rho^2 (t,x)dx - \int_0^L \frac{1}{2} \rho_0^2
(x)dx + \int_0^t \int_0^L \delta \rho_x^2 \,dx\,d\tau \\
&= \int_0^t \int_0^L \{
-a \rho u_x -b \rho \rho_x + c \rho \} \,dx\,d\tau
\end{aligned}
\end{equation}
Also, as $u \in \mathcal{V}_m$ we have, as a consequence of \eqref{eq7.3}b
$$
\int_0^L (u_t + b_2 u_x + RT_x + d\rho_x - fu_{xx} - g) udx =0 \,.
$$
If we then integrate this last result over $[0,t]$ we obtain
\begin{equation}\label{eq7.6}
\begin{aligned}
&\int_0^L \frac{1}{2} u^2 (t,x)dx - \int_0^L \frac{1}{2} u_0^2 (x)dx +
\int_0^t \int_0^L fu_x^2 \,dx\,d\tau \\
&= \int_0^t \int_0^L \{ -b_2 uu_x - RuT_x - du \rho_x + gu \} \,dx\,d\tau
\end{aligned}
\end{equation}
Next, as $T \in \mathcal{W}_m$, it follows from \eqref{eq7.3}c that
$$
\int_0^L (T_t + b_3 T_x + hu_x - iT_{xx} -j) Tdx =0
$$
and integrating this result over $[0,t]$ we obtain
\begin{equation}\label{eq7.7}
\begin{aligned}
&\int_0^L \frac{1}{2} T^2 (t,x)dx - \int_0^L \frac{1}{2} T_0^2 (x)dx +
\int_0^t \int_0^L i T_x^2 \,dx\,d\tau \\
&= \int_0^t \int_0^L \{ -b_3 TT_x - hTu_x + jT \} \,dx\,d\tau\,.
\end{aligned}
\end{equation}
Since $\rho_{xx} \in \mathcal{V}_m$, \eqref{eq7.3}a yields
$$
\int_0^L (\rho_t + au_x + b \rho_x - c - \delta \rho_{xx}) \rho_{xx} dx=0,
$$
and an integration by parts produces
\begin{equation}\label{eq7.8}
\rho_t \rho_x \big|_0^L - \int_0^L \rho_x \rho_{xt} dx
+ \int_0^L (au_x \rho_{xx} + b \rho_x \rho_{xx} - c \rho_{xx}
- \delta \rho^2_{xx})dx =0\,.
\end{equation}
Integrating by parts the first
three terms in last integral in \eqref{eq7.8} we obtain
\begin{align*}
&\int_0^L \Big( au_x \rho_{xx} + \frac{1}{2} b(\rho_x)_x^2 - c \rho_{xx}
\Big) dx \\
&= \Big( au_x \rho_x + \frac{1}{2} b \rho_x^2 - c\rho_x \Big)\Big|_0^L
- \int_0^L \Big( au_{xx} \rho_x + \frac{1}{2} b_x \rho^2_x - c_x
\rho_x \Big) dx \,.
\end{align*}
But $\rho_x (L,t)=0$, and $c(0,t) = c(L,t) =0$, so \eqref{eq7.8} and this
last identity yields
\begin{equation}\label{eq7.9}
\begin{aligned}
&\rho_t \rho_x \big|_0^L - \int_0^L \rho_x \rho_{xt} dx
+ \int_0^L \Big(-au_{xx} \rho_x - \frac{1}{2} b_x \rho_x^2 + c_x \rho_x
- \delta \rho^2_{xx}\Big) dx\\
&-a (0,t) u_x (0,t) \rho_x (0,t) - \frac{1}{2} b(0,t) \rho_x^2 = 0
\end{aligned}
\end{equation}
The first term on the left-hand side of \eqref{eq7.9} vanishes as
$\rho_x (L,t) = \rho_t (0,t) = 0$, for all $t \geq 0$, and this reduces
\eqref{eq7.9} to
\begin{align*}
\int_0^L \rho_x \rho_{xt} dx
&= -a(0,t) u_x (0,t) \rho_x (0,t) - \frac{1}{2} b(0,t)\rho_x^2 \\
&\quad + \int_0^L \Big( -au_{xx} \rho_x - \frac{1}{2} b_x \rho_x^2
+ c_x \rho_x - \delta \rho^2_{xx} \Big) dx
\end{align*}
which after integration over $[0,t]$ becomes the identity
\begin{equation}\label{eq7.10}
\begin{aligned}
&\int_0^L \frac{1}{2} \rho_x^2 (t,x)dx - \int_0^L \frac{1}{2} \rho_x^2
(x,0)dx + \int_0^t \int_0^L \delta \rho^2_{xx} \,dx\,d\tau + \frac{1}{2}
\int_0^t b(0,\tau) \rho_x (0, \tau)^2 d\tau
\\
&= -\int_0^t a(0, \tau) u_x (0,\tau) \rho_x (0, \tau) d\tau + \int_0^t
\int_0^L \Big( -au_{xx} \rho_x - \frac{1}{2} b_x \rho_x^2 + c_x \rho_x
\Big) \,dx\,d\tau\,.
\end{aligned}
\end{equation}
We note that, in \eqref{eq7.10}, we have $b(0,\tau) \geq \zeta >0$,
$0 \leq \tau \leq t$. As $u_{xx} \in \mathcal{V}_m$, \eqref{eq7.3}b yields
$$
\int_0^L (u_t + b_2 u_x + RT_x + d \rho_x - fu_{xx} - g) u_{xx} dx =0 \,.
$$
Integrating by parts in this last result, and using the fact that
$u_x (L,t)= u_t(0,t) = 0$, $t \geq 0$, yields
$$
\int_0^L u_x u_{xt} dx = \int_0^L (b_2 u_x + RT_x + d \rho_x - fu_{xx} -
g) u_{xx} dx
$$
which upon integration over $[0,t]$ produces the identity
\begin{equation}\label{eq7.11}
\begin{aligned}
&\int_0^L \frac{1}{2} u_x^2 (t,x)dx - \int_0^L \frac{1}{2} u^2_x (x,0)dx +
\int_0^t \int_0^L f u^2_{xx} \,dx\,d\tau \\
& = \int_0^t \int_0^L (b^2 u_x u_{xx} + RT_x u_{xx} + d \rho_x u_{xx} -
gu_{xx} )\,dx\,d\tau\,.
\end{aligned}
\end{equation}
Next, as $T_{xx} \in \mathcal{W}_m$, \eqref{eq7.3}c produces
$$
\int_0^L (T_t + b_3 T_x + hu_x - iT_{xx} - j) T_{xx} dx=0\,.
$$
In this last equation we integrate by parts, apply the conditions
$T_t (0,t) = T_t (L,t) = 0, t \geq 0$, and then integrate over $[0,t]$
so as to obtain
\begin{equation}\label{eq7.12}
\begin{aligned}
&\int_0^L \frac{1}{2} T_x^2 (t,x) dx - \int_0^L \frac{1}{2} T_x^2 (x,0)dx +
\int_0^t \int_0^L i T_{xx}^2 \,dx\,d\tau \\
&= \int_0^t \int_0^L (b_3 T_x T_{xx} + hu_x T_{xx} - j T_{xx}) \,dx\,d\tau
\end{aligned}
\end{equation}
Since $\rho_t \in \mathcal{V}_m$, \eqref{eq7.3}a yields
$$
\int_0^L (\rho_t + au_x + b \rho_x - c - \delta \rho_{xx}) \rho_t dx = 0\,;
$$
so that
\begin{equation}\label{eq7.13}
\int_0^t \int_0^L \rho^2_t \,dx\,d\tau = \int_0^t \int_0^L (-a \rho_t u_x - b
\rho_t \rho_x + c \rho_t + \delta \rho_t \rho_{xx}) \,dx\,d\tau\,.
\end{equation}
Next, we note that as $u_t \in \mathcal{V}_m$, \eqref{eq7.3}b yields
\begin{equation}\label{eq7.14}
\int_0^t \int_0^L (u_t + b_2 u_x + RT_x + d \rho_x - fu_{xx} - g) u_t \,dx\,d\tau = 0\,.
\end{equation}
Integrating the next to last term in this integral by parts we find that, as
$u_x (L,t) = u_t (0,t) = 0$,
$$
\int_0^t \int_0^L fu_{xx} u_t \,dx\,d\tau
= -\int_0^L \frac{1}{2} f u_x^2 dx\big|_{0}^{t}
+ \int_0^t \int_0^L \frac{1}{2} f_t u_x^2 \,dx\,d\tau - \int_0^t
\int_0^L f_x u_x u_t \,dx\,d\tau
$$
and if we now substitute this last result back into \eqref{eq7.14} we obtain
\begin{equation}\label{eq7.15}
\begin{aligned}
&\int_0^t \int_0^L u_t^2 \,dx\,d\tau + \int_0^L \frac{1}{2} fu_x^2 dx \\
&= \int_0^L \frac{1}{2} f(x,0) u_x^2 (x,0)dx \\
&\quad + \int_0^t \int_0^L \Big( -b_2 u_t u_x - Ru_t T_x - du_t \rho_x
- f_x u_t u_x + \frac{1}{2} f_t u_x^2 - gu_t \Big) \,dx\,d\tau\,.
\end{aligned}
\end{equation}
Using the fact that $T_t \in \mathcal{W}_m$ we obtain from \eqref{eq7.3}c
\begin{equation}\label{eq7.16}
\int_0^t \int_0^L (T_t + b_3 T_x + hu_x - i T_{xx} - j) T_t \,dx\,d\tau = 0\,.
\end{equation}
Integration of the next to the last term in \eqref{eq7.16} by parts, and use
of the conditions $T_t (L,t) = T_t (0,t) =0$, produces the identity
$$
\int_0^t \int_0^L iT_{xx} T_t \,dx\,d\tau = -\int_0^L \frac{1}{2} iT_x^2 dx
\big|_{0}^{t} + \int_0^t \int_0^L \frac{1}{2} i_t T_x^2 \,dx\,d\tau - \int_0^t
\int_0^L i_x T_x u_t \,dx\,d\tau
$$
which, when substituted in \eqref{eq7.16} yields
\begin{equation}\label{eq7.17}
\begin{aligned}
&\int_0^t \int_0^L T_t^2 \,dx\,d\tau + \int_0^L \frac{1}{2} iT_x^2 dx \\
&= \int_0^L \frac{1}{2} i(x,0) T_x^2 (x,0)dx\\
&\quad + \int_0^t \int_0^L \Big( -b_3 T_t T_x - hT_t u_x - i_x T_t T_x +
\frac{1}{2} i_t T_x^2 - j T_t \Big) \,dx\,d\tau\,.
\end{aligned}
\end{equation}
Adding together the results in \eqref{eq7.5}, \eqref{eq7.6}, \eqref{eq7.7},
\eqref{eq7.10}, \eqref{eq7.11}, \eqref{eq7.12}, \eqref{eq7.13},
\eqref{eq7.15}, and \eqref{eq7.17}, and then grouping like terms together,
we obtain the identity \eqref{eq7.4}.
\end{proof}
From the energy identity \eqref{eq7.4} we are now able to obtain for
$\hat{\mathcal{E}_{\delta}}(t)$ an energy inequality entirely analogous to
\eqref{eq6.35} for $\mathcal{E}_{\delta}(t)$.
\begin{lemma}\label{lem:7.2}
Under the conditions in Lemma \ref{lem:7.1}, we have for $\rho, u \in
\mathcal{V}_m, T \in \mathcal{W}_m$, and $0 < \delta < 1$,
\begin{equation}\label{eq7.18}
\hat{\mathcal{E}_{\delta}} (t) \leq \hat{\mathcal{E}}(0) + \mathcal{G}t +
\mathcal{K} \int_0^t \hat{\mathcal{E}_{\delta}} (\tau) d\tau
\end{equation}
for some positive constants $\mathcal{G}$ and $\mathcal{K}$, where
$\hat{\mathcal{E}_{\delta}} (0)$, being independent of $\delta$, has been
denoted as $\hat{\mathcal{E}}(0)$.
\end{lemma}
\noindent\textbf{Remarks:} \
(i) In view of the definition of
$\hat{\mathcal{E}_{\delta}} (t), \hat{\mathcal{E}}(0)$ is independent of
$\delta$.
(ii) The terms on the right-hand side of \eqref{eq7.4} have been
grouped by $\{ \ \}$ into four distinct subsets of terms; estimates for
typical terms in each of these four groupings are derived in the proof of
Lemma \ref{lem:7.2} and generic positive constants $C_i,K_i, G_i$
will be used in these estimates.
(iii) For $\delta$, satisfying $0 < \delta < 1$, a stronger result
than \eqref{eq7.18} actually follows from the proof of the Lemma, namely,
\begin{equation}\label{eq7.19}
\hat{\mathcal{E}_{\delta}}(t) + \mathcal{I} \leq \hat{\mathcal{E}} (0) +
\mathcal{G}t + \mathcal{K} \int_0^t \hat{\mathcal{E}} (\tau) d\tau
\end{equation}
with
\begin{equation}\label{eq7.20}
\mathcal{I} = \frac{1}{2} \int_0^t \int_0^L \delta \rho^2_{xx} \,dx\,d\tau
\end{equation}
\begin{proof}[Proof of Lemma \ref{lem:7.2}]
The terms in the first grouping on the
right-hand side of \eqref{eq7.4} involve functions and/or first derivatives
of functions with bounded coefficients; these may be estimated as in the
following sample case:
\begin{equation}\label{eq7.21}
\Big| \int_0^t \int_0^L a \rho u_x \,dx\,d\tau \Big|
\leq \frac{\sup |a|}{2}
\int_0^t \int_0^L (\rho^2 + u_x^2) \,dx\,d\tau = K_1 \int_0^t \int_0^L (\rho^2 +
u_x^2) \,dx\,d\tau\,.
\end{equation}
The second group of terms involves second derivatives or time derivatives of
the functions $\rho, u$, and $T$. These can be estimated as follows: for any
$\eta >0$,
\begin{subequations}\label{eq7.22}
\begin{gather}
\Big| \int_0^t \int_0^L Ru_{xx} T_x \,dx\,d\tau \Big|
\leq \eta R \int_0^t
\int_0^L u_{xx}^2 \,dx\,d\tau + \frac{R}{4\eta}
\int_0^t \int_0^L T_x^2 \,dx\,d\tau,
\\
\Big| \int_0^t \int_0^L Ru_{xx} T_x \,dx\,d\tau \Big|
\leq \eta C_1 \int_0^t \int_0^L u_{xx}^2 \,dx\,d\tau + K_2 (\eta)
\int_0^t \int_0^L T_x^2 \,dx\,d\tau
\end{gather}
where $K_2(\eta)$ indicates that the constant $K_2$ depends on
the choice of the parameter $\eta$.
One further example of this type would be
\begin{equation}
\begin{aligned}
\Big| \int_0^t \int_0^L a \rho_t u_x \,dx\,d\tau \Big|
&\leq \eta \sup |a| \int_0^t \int_0^L \rho_t^2 \,dx\,d\tau
+ \frac{\sup |a|}{4\eta} \int_0^t \int_0^L u_x^2 \,dx\,d\tau \\
&= \eta C_2 \int_0^t \int_0^L \rho_t^2 \,dx\,d\tau + K_4 (\eta)
\int_0^t \int_0^L u_x^2 \,dx\,d\tau\,.
\end{aligned}
\end{equation}
\end{subequations}
The estimates for the third group of terms are similar to those in the
following two examples:
\begin{subequations}\label{eq7.23}
\begin{gather}
\begin{aligned}
\Big| \int_0^t \int_0^L c \rho \,dx\,d\tau \Big|
&\leq \frac{1}{2} \int_0^t \int_0^L \rho^2 \,dx\,d\tau
+ \frac{1}{2} \int_0^t \int_0^L c^2 \,dx\,d\tau \\
&\leq K_3 \int_0^t \int_0^L \rho^2 \,dx\,d\tau + G_1 \cdot t\,,
\end{aligned}\\
\begin{aligned}
\Big| \int_0^t \int_0^L gu_{xx} \,dx\,d\tau \Big|
&\leq \eta \int_0^t \int_0^L
u_{xx}^2 \,dx\,d\tau + \frac{1}{4\eta} \int_0^t \int_0^L g^2 \,dx\,d\tau \\
&\leq \eta \int_0^t \int_0^L u_{xx}^2 \,dx\,d\tau + \frac{{\rm const.}}{4\eta} \cdot t
\\
&= \eta C_4 \int_0^t \int_0^L u_{xx}^2 \,dx\,d\tau + G_2 (\eta) \cdot t\,.
\end{aligned}
\end{gather}
\end{subequations}
Finally, the last term on the right-hand side of \eqref{eq7.4}; i.e.,
$ \int_0^t \int_0^L \delta \rho_t \rho_{xx} \,dx\,d\tau$ is
estimated exactly as in \eqref{eq6.41}, which we rewrite here as
\begin{equation}\label{eq7.24}
\Big| \int_0^t \int_0^L \delta \rho_t \rho_{xx} \,dx\,d\tau \Big|
\leq \int_0^t \int_0^L \frac{\delta}{2} \rho_t^2 \,dx\,d\tau + \mathcal{I}\,.
\end{equation}
Now, terms which appear on the right-hand sides of estimates such as
\eqref{eq7.22}a,b,c, \eqref{eq7.23}b, et.al., and which are multiplied by
$\eta$, may be absorbed by the similar terms on the left-hand side of
\eqref{eq7.4} because of \eqref{eq6.9}a,b; in particular, for $\eta$
sufficiently small,
\begin{subequations}\label{eq7.25}
\begin{gather}
\int_0^t \int_0^L (i_c - C\eta) T_{xx}^2 \,dx\,d\tau
\geq \int_0^t \int_0^L \frac{1}{2} i_c T_{xx}^2 \,dx\,d\tau \\
\int_0^t \int_0^L (f_c - C\eta) u_{xx}^2 \,dx\,d\tau \geq \int_0^t \int_0^L
\frac{1}{2} f_c u_{xx}^2 \,dx\,d\tau \\
\int_0^t \int_0^L (1 - C\eta) T_t^2 \,dx\,d\tau \geq \int_0^t \int_0^L
\frac{1}{2} T_t^2 \,dx\,d\tau \\
\int_0^t \int_0^L (1 - C\eta) u_t^2 \,dx\,d\tau \geq \int_0^t \int_0^L
\frac{1}{2} u_t^2 \,dx\,d\tau \\
\int_0^t \int_0^L \Big( \frac{1}{2} - C\eta \Big) \rho_t^2 \,dx\,d\tau
\geq \int_0^t \int_0^L \frac{1}{4} \rho_t^2 \,dx\,d\tau\,,
\end{gather}
\end{subequations}
where $C = \sum C_i$. Note that once $\eta$ is chosen, the $K_i$ and $G_i$,
$i = 1, 2, \ldots$ are constants.
In \eqref{eq7.24}, the term
\begin{equation}\label{eq7.26}
\mathcal{J} = \frac{1}{2} \delta \int_0^t \int_0^L \rho_t^2 \,dx\,d\tau
\end{equation}
may be absorbed by the similar term on the left-hand side of \eqref{eq7.4}
and, as $0 < \delta <1$, we will have $1-\frac{\delta}{2} > \frac{1}{2}$.
Also, the integral $\mathcal{I}$, on the right-hand side of \eqref{eq7.24}
will be absorbed by the term
$$
\delta \int_0^t \int_0^L \rho^2_{xx} \,dx\,d\tau \equiv 2\mathcal{I}
$$
on the left-hand side of \eqref{eq7.4} leaving a balance of $\mathcal{I}$
among the terms on the left-hand side of \eqref{eq7.4}. After all the
above-referenced terms are absorbed (on the left-hand side of \eqref{eq7.4})
we see that as a lower bound for the left-hand side of \eqref{eq7.4} we have
the expression
\begin{subequations}\label{eq7.27}
\begin{equation}
\hat{\mathcal{E}_{\delta}} (t) + \mathcal{I} + \frac{1}{2} \int_0^t b(0,t)
\rho_x (0,\tau)^2 d\tau
\end{equation}
while, adding all the estimates referenced above, yields an upper
bound for the remaining terms on the right-hand side of \eqref{eq7.4} of
the form
\begin{equation}
\hat{\mathcal{E}}(0) + \mathcal{G}t + \mathcal{K} \int_0^t
\hat{\mathcal{E}_{\delta}} (\tau) d\tau\,,
\end{equation}
\end{subequations}
where $\mathcal{K} = \sum K_i$ and $\mathcal{G} = \sum G_i$. Combining the
lower bound for the left-hand side of \eqref{eq7.4}; i.e., \eqref{eq7.27}a,
with the upper bound for the surviving terms on the right-hand side of
\eqref{eq7.4}, and taking note of the fact that $b(0,t) \geq \zeta >0$ in
\eqref{eq7.27}a, we are led to the estimate \eqref{eq7.19}.
\end{proof}
From \eqref{eq7.18} and Gronwall's inequality we may now conclude, as in
Lemma \ref{lem:6.3}, the following result.
\begin{lemma}\label{lem:7.3}
Under the conditions stated in Lemma \ref{lem:7.1} we have for $\rho, u \in
\mathcal{V}_m, T \in \mathcal{W}_m$, $0 < \delta < 1$, and all
$t, 0 \leq t \leq t_0$, that there exists $\hat{\mathcal{C}_{t_0}} > 0$
such that
\begin{equation}\label{eq7.28}
\hat{\mathcal{E}_{\delta}} (t) \leq \hat{\mathcal{C}_{t_0}} \Big(
\frac{\mathcal{G}}{\mathcal{K}} + \hat{\mathcal{E}} (0) \Big)
\end{equation}
\end{lemma}
To obtain the required existence and uniqueness result for the
regularized problem consisting of \eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c,
\eqref{eq6.8}a-f, we proceed exactly as in \S6, namely, (i) we introduce the
Galerkin approximations \eqref{eq6.52}a,b,c with coefficients $A_{lm},
B_{lm}, C_{lm}$ satisfying \eqref{eq6.53}a,b,c, where $\zeta_l, \eta_l,
\gamma_l$ satisfy \eqref{eq6.54}a,b,c, (ii) we require that the coefficients
in the Galerkin approximations satisfy the coupled system of ordinary
differential equations \eqref{eq6.55}a,b,c, (iii) we invoke the result of
Lemma \ref{lem:6.4} and show that the approximate solutions $\rho_m, u_m,
T_m$ defined by \eqref{eq6.52}a,b,c satisfy the hypotheses of Lemma
\ref{lem:7.1} and \ref{lem:7.2}, and (iv) using the estimate for $(\rho_m,
u_m, T_m)$; i.e., on any interval $[0,t_0]$
\begin{equation}\label{eq7.29}
\hat{\mathcal{E}}_{\delta m} (t) \leq \hat{\mathcal{C}}_{t_0} \Big(
\frac{\mathcal{G}}{\mathcal{K}} + \hat{\mathcal{E}}_m (0) \Big)\,,
\end{equation}
we conclude that $\rho_m, u_m, T_m$ satisfy \eqref{eq6.68}a,b for $0 \leq t
\leq t_0$. The remaining parts of the proof of Theorem \ref{theor:6.1}
remain unchanged in the present circumstances and, thus, we are able to
conclude, once again, that the regularized initial-boundary value problem
\eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c, \eqref{eq6.8}a-f has, for each
$\delta >0$, a unique
solution $(\rho^{\delta}, u^{\delta}, T^{\delta})$, for any $t>0$,
such that \eqref{eq6.66}a,b are
satisfied; furthermore, the higher regularity result expressed by Theorem
\ref{theor:6.2}; i.e., \eqref{eq6.81}, also holds in the present
circumstances in which we have imposed the hypothesis \eqref{eq7.1}a,b.
As the Galerkin approximations $\rho_m, u_m, T_m$,
converge in
$W^{1,2} ([0,t]; L^2 [0,L]) \cap L^2 ([0,t]; W^{2,2} [0,L])$
to the unique solution of \eqref{eq6.10}a,b,c,
\eqref{eq6.7}a,b,c, \eqref{eq6.8}a-f, we have for the limit $\rho^{\delta},
u^{\delta}, T^{\delta}$ of these sequences the estimate
\begin{equation}\label{eq7.30}
\hat{\mathcal{E}}_{\delta} (t) \leq \hat{\mathcal{C}}_{t_0} \Big(
\frac{\mathcal{G}}{\mathcal{K}} + \hat{\mathcal{E}} (0) \Big) \equiv
\mathcal{C}'_{t_0}
\end{equation}
for all $\delta >0$ and $t \in (0, t_0)$. Therefore, the solution set
$(\rho^{\delta}, u^{\delta}, T^{\delta})$ for the problem
\eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c, \eqref{eq6.8}a-f satisfies the
following estimates:
\begin{subequations}\label{eq7.31}
\begin{gather}
\| \rho^{\delta} \|^2_{L^2([0,t]; W^{1,2}([0,L]))} + \| \rho^{\delta}_{,t}
\|^2_{L^2 ([0,t]; L^2([0,L]))} \leq \mathcal{C}_1 \,,\\
\|u^{\delta} \|^2_{L^2([0,t]; W^{2,2} ([0,L]))} + \| u^{\delta}_{,t}
\|^2_{L^2([0,t]; L^2([0,L]))} \leq \mathcal{C}_2 \,, \\
\| T^{\delta} \|^2_{L^2([0,t]; W^{2,2}([0,L]))} + \| T^{\delta}_{,t}
\|^2_{L^2([0,t]; L^2([0,L]))} \leq \mathcal{C}_3
\end{gather}
\end{subequations}
for positive generic constants $\mathcal{C}_i$, $i=1,2,3$, which are
independent of $\delta$. From \eqref{eq7.31}a,b,c it follows that there
exists a triplet $(\rho, u, T)$ with
\begin{subequations}\label{eq7.32}
\begin{gather}
\rho \in L^2 ([0,t]; W^{1,2} ([0,L]), \quad \rho_t \in L^2 ([0,t]; L^2([0,L]))\,,
\\
u \in L^2 ([0,t]; W^{2,2} ([0,L])), \quad u_t \in L^2 ([0,t]; L^2 ([0,L]))\,, \\
T \in L^2([0,t]; W^{2,2} ([0,L])), \quad T_t \in L^2 ([0,t]; L^2([0,L]))
\end{gather}
\end{subequations}
and a sequence $(\rho^{\delta_k}, u^{\delta_k}, T^{\delta_k})$ of solutions
to the problem \eqref{eq6.10}a,b,c, \eqref{eq6.7}a,b,c, \eqref{eq6.8}a-f
such that
\begin{subequations}\label{eq7.33}
\begin{gather}
\rho^{\delta_k} \rightharpoonup \rho \quad \text{weakly in } L^2 ([0,t];
W^{1,2} ([0,L])) \\
\rho^{\delta_k}_{,t} \rightharpoonup \rho_t \quad \text{weakly in }
L^2([0,t]; L^2 ([0,L]))
\end{gather}
\end{subequations}
\begin{subequations}\label{eq7.34}
\begin{gather}
u^{\delta_k} \rightharpoonup u \quad \text{weakly in } L^2([0,t]; W^{2,2}
([0,L]) \,,\\
u^{\delta_k}_{,t} \rightharpoonup u_t \quad \text{weakly in } L^2 ([0,t];
L^2 ([0,L])
\end{gather}
\end{subequations}
and
\begin{subequations}\label{eq7.35}
\begin{gather}
T^{\delta_k} \rightharpoonup T \quad \text{weakly in } L^2([0,t]; W^{2,2}
([0,L])\,, \\
T^{\delta_k}_{,t} \rightharpoonup T_{,t} \quad \text{weakly in } L^2 ([0,t]; L^2
([0,L])\,.
\end{gather}
\end{subequations}
However, the triplet $(\rho^{\delta_k}, u^{\delta_k}, T^{\delta_k})$,
satisfies \eqref{eq6.74}a, \eqref{eq6.74}b, \eqref{eq6.74}c, with
$\delta = \delta^k$ for any $\psi \in L^2([0,t]; \mathcal{C}_0^{\infty}([0,L]))$;
letting $\delta^k \to 0$ we conclude that
\begin{equation}\label{eq7.36}
(\rho, u, T) = \lim_{\delta^k \to 0} (\rho^{\delta_k}, u^{\delta_k},
T^{\delta_k})
\end{equation}
satisfies
\begin{subequations}\label{eq7.37}
\begin{gather}
\int_0^t \int_0^L (\rho_t + a u_x + b \rho_x - c) \psi \,dx\,d\tau = 0 \,,\\
\int_0^t \int_0^L (u_t + b_2 u_x + RT_x + d \rho_x - fu_{xx} - g) \psi
\,dx\,d\tau = 0 \,,\\
\int_0^t \int_0^L (T_t + b_3 T_x + hu_x - i T_{xx} - j)\psi \,dx\,d\tau = 0
\end{gather}
\end{subequations}
for all $\psi \in L^2 ([0,t]; \mathcal{C}_0^{\infty} ([0,L])$. Thus,
$(\rho, u, T)$ is a weak solution of the problem \eqref{eq6.6}a,b,c. Furthermore, by
virtue of the usual trace theorem and \eqref{eq7.31}a,b,c we can conclude
that $(\rho, u, T)$ also satisfies the initial conditions \eqref{eq6.7}a,b,c
as well as the boundary conditions \eqref{eq6.8}a-e.
The argument delineated above has established the following result.
\begin{theorem}\label{theor:7.1}
Given the hypotheses \eqref{eq6.8}a,b,c and \eqref{eq7.1}a,b, the mixed
hyperbolic-parabolic initial boundary-value problem \eqref{eq6.6}a,b,c,
\eqref{eq6.7}a,b,c, \eqref{eq6.8}a-e has a unique solution $(\rho, u, T)$
which satisfies \eqref{eq7.32}a,b,c.
\end{theorem}
\begin{figure}[ht]
\begin{center} \scriptsize
\setlength{\unitlength}{1mm}
\begin{picture}(105,55)(0,0)
\put(0,0){\includegraphics[width=0.8\textwidth]{fig1}}
\put(17,53){Combustor Cycle}
\put(73,53){Acoustic Pressure}
\put(13,45){Spark plug}
\put(30,43){Combustion}
\put(30,40){Chamber}
\put(-1,39){Fuel}
\put(0,36){+}
\put(-1,33){Air}
\put(2,27){Flapper}
\put(2,24){Valve}
\put(32,18){Flue Gas}
\put(66,17){0}
%\put(55,42){Pressure}
\put(76,16){Cycle Time}
\put(99,17){1}
\put(4,29.5){Starting Cycle (Inlet Valves Open)}
\put(4,14.5){Positive Pressure Cycle (Inlet Valves Closed)}
\put(4,0){Negative Pressure Cycle (Inlet Valves Open)}
\put(32,32.5){tailpipe}
\put(-1,11){Fuel}
\put(0,8){+}
\put(-1,5){Air}
\end{picture}
\end{center}
\caption{The limit of the integrand does not exist} \label{fig1}
\end{figure}
\begin{figure}[ht]
\begin{center} \scriptsize
\setlength{\unitlength}{1mm}
\begin{picture}(105,50)(0,0)
\put(0,0){\includegraphics[width=0.8\textwidth]{fig2}}
\put(44,48){Gas Valve}
\put(44,45){Flow Area, $A_g$}
\put(72,46){Equivalent plane flame}
\put(72,43){with area, $A_B$, and}
\put(72,40){speed, $U_f$}
\put(75,33){Flow Area, $A$}
\put(75,26){Exhaust Pipe}
\put(100,22){$p_a $}
\put(80,17){$L $}
\put(33,33){Pressure, $p$}
\put(33,26){Reactants}
\put(52,22.5){Combustion}
\put(52,20.3){Products}
\put(41,19.5){$U_f$}
\put(33,12){Chamber}
\put(33,9){Volume $V_B$}
\put(33,3){Combustion}
\put(33,0){Chamber}
\put(10,31){$p_g,T_g$}
\put(10,28){Gas Supply}
\put(10,20){Air Supply}
\put(10,17){$p_0,T_0$}
\put(0,5){Air valve}
\put(0,2){flow area, $A_0$}
\end{picture}
\end{center}
\caption{Parameters in the Pulse Combustor Model}
\label{fig2}
\end{figure}
\begin{figure}[ht]
\begin{center} \scriptsize
\setlength{\unitlength}{1mm}
\begin{picture}(100,36)(0,0)
\put(0,0){\includegraphics[width=0.8\textwidth]{fig3}}
\put(41,18){$L_1$}
\put(57,18){$L_2$}
\put(98,18){$L_3$}
\put(99,13){$x$}
\put(0,15){$O$}
\end{picture}
\end{center}
\caption{Pulse combustor configuration}
\label{fig3}
\end{figure}
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\end{document}