\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 70, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/70\hfil One-dimensional adhesion model]
{One-dimensional adhesion model for large scale
structures}
\author[K. T. Joseph \hfil EJDE-2010/70\hfilneg]
{Kayyunnapara Thomas Joseph}
\dedicatory{Dedicated to the memory of Professor P. L. Sachdev}
\address{Kayyunnapara Thomas Joseph \newline
School of Mathematics\\
Tata Institute of Fundamental Research\\
Homi Bhabha Road\\
Mumbai 400005, India}
\email{ktj@math.tifr.res.in}
\thanks{Submitted January 30, 2010. Published May 17, 2010.}
\subjclass[2000]{35A20, 35L50, 35R05}
\keywords{Adhesion approximation; large scale structure}
\begin{abstract}
We discuss initial value problems and initial boundary value problems
for some systems of partial differential equations appearing in the
modelling for the large scale structure formation in the universe.
We restrict the initial data to be bounded measurable and locally
bounded variation function and use Volpert product to justify the
product which appear in the equation. For more general initial data
in the class of generalized functions of Colombeau, we construct
the solution in the sense of association.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
The Burgers equation in one dimension is a second-order nonlinear
parabolic equation
balancing quadratic nonlinearity and diffusion and is of the form
\begin{equation}
\begin{aligned}
u_t + u u_x ={\epsilon} u_{xx}.
\end{aligned}
\label{e1.1}
\end{equation}
This equation was introduced by Burgers in 1939
as a simplification of the Navier-Stokes equation with the hope of
understanding issues such as turbulence. Hopf \cite{h1} and Cole
\cite{col1}
showed that Burgers equation can be integrated explicitly. Indeed
Burgers equation does not posses the fundamental aspects of turbulence,
namely sensitivity to small perturbations in the initial
conditions and the spontaneous arise of randomness by chaotic dynamics.
Hopf's construction
of solution of the inviscid Burgers equation, by passing to the limit
as $\epsilon$ goes to zero in the explicit solution of \eqref{e1.1} with
with initial data $u(x,0)=u_0(x)$ in the space of bounded measurable
function and the
work of Lax \cite{la1} for systems laid the mathematical foundation of
the theory of Hyperbolic systems of conservation laws.
In the 1980's Burgers equation in one dimensions and in
multi-dimensions again started to appear in other fields
such as in statistical mechanics and in cosmology and became focus of
study again.
In this paper we study Burgers equation which has
applications in cosmology where it is closely linked to what is usually
referred to as Zeldovich approximation \cite{z1}. According to this
model the evolution in the last stage of the expansion of the
universe, matter is described as cold dust moving under gravity alone
and the laws are governed by the system
\begin{equation}
\begin{gathered}
u_t + (u.\nabla)u =0,\quad u=\nabla \phi\\
\rho_t + \nabla(\rho u)=0,
\end{gathered}\label{e1.2}
\end{equation}
where $u$ is velocity and $\rho$ the density of the particles.
It is observed that the fastest growing mode in linear theory has
decaying vorticity, and this is the reason for the interest is in
potential solutions
where the velocity $u$ can be represented in terms of a velocity
potential $\phi$.
Even with smooth initial data
\begin{equation}
u(x,0)=u_0(x),\quad \rho(x,0)=\rho_0(x)\label{e1.3}
\end{equation}
in general, existence of global smooth solutions is not possible as the
fastest particles overrun the slowest ones and
after some time the density becomes infinite.
Mathematically, the local existence theory given by classical Hamilton
Jacobi theory of first order equation is not helpful to construct global
solution. The mapping from Lagrangian space$ L(y)$ to Eulerian space
$E(x)$ given by
\begin{equation}
x = y + t u_0(y) \label{e1.4}
\end{equation}
is one-one only for short time and
\begin{equation}
u(x,t)=u_0(y),\quad
\rho(x,t)=\frac{\rho_0(y)}{\det(\frac{\partial x_i}{\partial y_j})}
\label{e1.5}
\end{equation}
gives only short time existence result.
Gurbatov and Saichev \cite{g1}, introduced the adhesion model.
In adhesion approximation, the motion of particles is described
by the motion of sticking particles. The velocity obey the Burgers
equation and density by the continuity equation.
\begin{equation}
\begin{gathered}
u_t + (u.\nabla)u =\epsilon \Delta u,\quad u=\nabla \phi\\
\rho_t + \nabla(\rho u)=0
\end{gathered}\label{e1.6}
\end{equation}
Using a Hopf-Cole transformation Weinberg and Gunn \cite{w1}, Joseph and
Sachdev \cite{j6}, wrote down the exact formula for the velocity
$u^\epsilon$ when the initial data is of the form
\[
u(x,0)=\nabla \phi_0(x)
\]
and derived a formula for the vanishing viscosity limit, generalizing
Hopf's \cite{h1} result, namely
\[
\lim_{\epsilon \to 0}u^\epsilon(x,t)=u(x,t)=
\frac{(x-y(x,t))}{t}
\]
where $y(x,t)$ is a minimizer in
\[
\min_{y\in R^n} \big\{\phi_0(y) + \frac{|x-y|^2}{t}\big\}
\]
which exists always and unique for almost every points. Joseph and
Sachdev \cite{j6} also studied the large time behaviour of the solution
for each fixed $\epsilon>0$. The velocity thus obtained is not
smooth and density remains to be determined.
Gurbatov and Saichev \cite{g2} proposed a model equation to
approximate the
adhesion model, to compute density. In the limit $\epsilon$
goes to 0, $y(x,t)$ is interpreted as Lagrangian co-ordinates of
particles falling into $x$ at time $t$. The matter density
$\rho$ is determined by Euler to Lagrange co-ordinate
transformation Jacobian given by \eqref{e1.5}.
The idea is formally introduce
\begin{equation}
x = y + t u(x,t)\label{e1.7}
\end{equation}
where $u$ satisfy the Burgers equation. Interpret this as
the mapping from Lagrangian space$ L(y)$ to Eulerian space
$E(x)$ and the density be defined
\[
\rho(x,t)=\rho_0 \quad \det(\frac{\partial y_i}{\partial x_j}).
\]
It was shown in \cite{g2} that $y=y(x,t)$ defined by \eqref{e1.7}
satisfies the equation
\[
y_t + (u.\nabla)y =\epsilon \Delta y,
\]
and in one dimension $\rho$ satisfies
\begin{equation}
\rho_t +(u \rho)_x =\epsilon \rho_{xx},
\label{e1.8}
\end{equation}
where $\rho_0$ is a constant.
For one space dimension, \eqref{e1.1} together with \eqref{e1.8}
is called the modified adhesion model.
Joseph \cite{j2} has shown that this modified adhesion model can be
linearized using a generalized Hopf-Cole transformation,
and showed the formation of $\delta$-waves in the density components
in the vanishing viscosity limit when the initial data is of Riemann
type.
The aim of this paper is to study one dimensional case,
and give simpler proofs for known results and give some new results on
initial
value problems and initial boundary value
problems. The focus is on the exact limit, its
structure and how to make sense of solution.
\section{Initial value problem}
In one space dimension the Zeldovich approximation takes the form
\begin{equation}
\begin{gathered}
u_t+(u^2/2)_x=0,\\
\rho_t +(\rho u)_x =0.
\end{gathered}\label{e2.1}
\end{equation}
With initial data
\begin{equation}
u(x,0)=u_0(x),\quad \rho(x,0)=\rho_0(x)\label{e2.2}
\end{equation}
Equation \eqref{e2.1} has a local smooth solution if the initial
data is smooth. The equation for $u$ says that
$u$ is constant along the characteristics given by \eqref{e1.4}. The
construction given by \eqref{e1.4}-\eqref{e1.5} is not valid when the
characteristics cross, the solution is multi-valued and shocks develop.
In this section we take the initial data \eqref{e2.2} and
use the model \eqref{e1.1} and \eqref{e1.8} to
construct explicit formula for the solution of \eqref{e2.1} and
\eqref{e2.2} as $\epsilon$ goes
to 0. For the case the initial data in $L^\infty \cup BV_{\rm loc}$, we show that
this explicit
formula for the inviscid problem is same as the one obtained in
LeFloch \cite{le1} and in Tan et al. \cite{ta1} by different methods.
The Riemann problem was treated earlier by Joseph \cite{j2}.
The solution shows that
the particles move with constant velocities until they collide and at
collision, the colliding particle form a new massive particle;
$u$ remains bounded pointwise and $\rho$ is not bounded pointwise but
are measures. Also there appear vacuum. We use the Volpert product
\cite{v1} to justify the product $\rho u$ appearing in the equation. Note
that the use of Volpert product was first advocated for hyperbolic
problems by LeFloch \cite{le2} and this provides only one particular
approach among a family of definitions described in Dal Maso, LeFloch
and Murat \cite{d1}.
Next we analyze the modified adhension model in the light of Colombeau
theory \cite{co1, co2, co3}. We
consider initial datas in more general class and construct
solution in the algebra of generalized functions of Colombeau \cite{co1}.
\subsection{Explicit formula for the modified adhesion model}
The modified adhesion model see Gurbatov and Saichev \cite{g2}
in one space dimension is
\begin{equation}
\begin{gathered}
u_t+(u^2/2)_x=\frac{\epsilon}{2} u_{xx},\\
\rho_t +(\rho u)_x =\frac{\epsilon}{2}\rho_{xx}.
\end{gathered} \label{e2.3}
\end{equation}
We solve \eqref{e2.3} with initial data
\begin{equation}
u(x,0)=u_0(x),\quad \rho(x,0)=\rho_0(x)\label{e2.4}
\end{equation}
using a generalized Hopf-Cole transformation see Joseph \cite{j2},
namely
\begin{equation}
u =-\epsilon \frac{a_x}{a}, \,\,\,\rho=(\frac{b}{a})_x
\label{e2.5}
\end{equation}
the problem \eqref{e2.3} and \eqref{e2.4} is reduced to the linear
problem
\begin{equation}
\begin{gathered}
a_t=\frac{\epsilon}{2} a_{xx},\quad
a(x,0) = e^{-\frac{U_0(x)}{\epsilon}}\\
b_t =\frac{\epsilon}{2}b_{xx},\quad
b(x,0)= R_0(x) e^{-\frac{U_0(x)}{\epsilon}}\\
R_0(x) =\int_0^x \rho(y) dy,\quad
U_0(x)=\int_0^x u_0(y) dy
\end{gathered} \label{e2.6}
\end{equation}
Solving \eqref{e2.6} and substituting in \eqref{e2.5}, we have
the following theorem.
\begin{theorem} \label{thm2.1}
Assume $u_0$ and $v_0$ are bounded measurable or integrable. Then
\begin{equation}
\begin{gathered}
u^\epsilon(x,t)=\int_{R^1} \frac{(x-y)}{t}
d\mu_{(x,t)}^{\epsilon}(y)\\
\rho^\epsilon(x,t)=\partial_{x}R^{\epsilon}(x,t)\\
R^{\epsilon}(x,t)=\int_{R^1} R_0(y)
d\mu_{(x,t)}^{\epsilon}(y).
\end{gathered} \label{e2.7}
\end{equation}
where for each $(x,t)$, and $\epsilon>0$, the probability measure
$d\mu_{(x,t)}^{\epsilon}(y)$ defined by
\begin{equation}
\begin{gathered}
d\mu_{(x,t)}^{\epsilon}(y) =
\frac{e^{-\frac{\theta(x,y,t)}{\epsilon}} dy}{\int_{R^1}
e^{-\frac{\theta(x,y,t)}{\epsilon}} dy}\\
\theta(x,y,t) = U_0(y)+\frac{(x-y)^2}{2t}.
\end{gathered} \label{e2.8}
\end{equation}
is a solution to \eqref{e2.3} and \eqref{e2.4}.
\end{theorem}
\subsection{Vanishing viscosity limit}
In this section, we find an explicit formula for global solution of
\eqref{e2.1} with initial data \eqref{e2.2} in the space of bounded
measurable functions which are locally BV functions. Here we use
the modified adhesion model \eqref{e2.3} and follow the
analysis of Hopf \cite{h1} and Lax \cite{la1} and the properties of the
minimizers of
\begin{equation}
\min_{-\infty0$, except for a countable $x$,
there exits a unique minimizer $y(x,t)$ for \eqref{e2.9} and
at these points
\begin{gather}
u(x,t)= \lim_{\epsilon \to 0} u^\epsilon(x,t)=
\frac{(x-y(x,t))}{t}, \label{e2.10}
\\
R(x,t)= \lim_{\epsilon \to 0}R^\epsilon(x,t)=
\int_0^{y(x,t)} \rho_0(z)dz.
\label{e2.11}
\end{gather}
The functions $u(x,t)$ and $R(x,t)$ are well defined a.e. and are
functions of bounded variation.
Further for each $t>0$, and $x\in R^1$, $u(x+,t)$ and $u(x-,t)$
$R(x-,t)$ and $R(x+,t)$ exits.
Also $u(x,t)$ satisfies the entropy condition $u(x-,t)\geq u(x+,t)$.
Finally
\begin{equation}
\rho(x,t) = \lim_{\epsilon \to 0}\rho^\epsilon(x,t)=\partial_{x}
(\int_0^{y(x,t)} \rho_0(z)dz)
\label{e2.12}
\end{equation}
in the sense of distributions.
Further, $(u,\rho)$ satisfies \eqref{e2.1} in the sense of distribution
and satisfies the initial conditions \eqref{e2.2}.
\end{theorem}
\begin{proof}
For initial data $u_0$ integrable or bounded measurable, one get that for
each fixed $(x,t)$, $\theta(x,y,t)$ has a global minimum as a
function of $y$ and minimum is achieved at some point. Let $y(x,t)$
be a point where global minimum is achieved in \eqref{e2.8}.
This minimizing point may not be unique.
Hopf \cite{h1} and Lax \cite{la1} analysis shows that he largest and the
smallest of these minimizers $y^{+}(x,t)$ and $y^{-}(x,t)$ are
increasing functions of $x$ and hence the point of discontinuities are
at most countable and except these points they are equal
$y(x,t)=y^{-}(x,t)=y^{+}(x,t)$. At these points where $y(x,t)$ is unique
the measure $d\mu_(x,t)^{\epsilon}(y) \to \delta_{y(x,t)}$
as $\epsilon$ goes to 0, in measure in the sense that for
any continuous function $g(y)$ on $R^1$
\[
\int g(y) d\mu_{(x,t)}^{\epsilon}(y)\to \langle
\delta_{y(x,t)},g(y)\rangle.
\]
We immediately get the limit \eqref{e2.10} and \eqref{e2.11}.
To show $(u,\rho)$ satisfies \eqref{e2.1}, we should interpret the
product $u \rho$ suitably as we only know that $u$ and $R$ are functions
of bounded variation and hence $\rho=R_x$ is a Radon measure.
We follow Volpert \cite{v1}.
With respect to
$u$, a function of bounded variation, we have the decomposition of
the domain $R^1\times[0, \infty)$
\[
[0,\infty)\times[0, \infty) =S_c \cup S_j \cup S_0
\]
where $S_c$ and $S_j$ are points of approximate continuity of $u$ and
points of approximate jump of $u$ and $S_0$ is a set of one dimensional
Hausdorff-measure zero. At any point $(x,t) \in S_j$, $u(x-0,t)$ and
$u(x+0,t)$ denote the left and right values of $u(x,t)$. For any
continuous function $g :R^1 \to R^1$, the Volpert product
$g(u)\rho =g(u)R_x$ is defined as a Borel measure in the following
manner.
Consider the averaged superposition of $g(u)$ (see Volpert \cite{v1})
\begin{equation}
\overline{g(u)}(x,t) = \begin{cases}
g(u(x,t),&\text{if } (x,t) \in S_c,\\
\int_0^1 g((1-\alpha)(u(x-,t)+\alpha u(x+,t))d\alpha,
&\text{if }(x,t) \in S_j .
\end{cases} \label{e2.13}
\end{equation}
Volpert \cite{v1} proves that $\overline{g(u)}$ is measurable and
locally integrable with respect to the Borel measure $R_x$, so that the
nonconservative product $\overline{g(u)}R_x$ has a meaning as a
locally finite Borel measure. In deed
\[
[g(u)R_x](A)=\int_{A}\overline{g(u)}(x,t)R_x
\]
where $A$ is a Borel measurable subset of $S_c$ and
\[
[g(u)R_x](\{(x,t)\})=\overline{g(u)}(x,t)(R(x+0,t)-R(x-0,t))
\]
provided $(x,t) \in S_j$.
To show $(u,\rho)$ is a solution, we need to show
\begin{equation}
(u, \phi_t) +\Big(\frac{(u^2)}{2},\phi_x\Big)=0,\quad
(\rho,\phi_t)+({\overline u}\rho, \phi_x)=0
\label{e2.14}
\end{equation}
for all test functions $\phi$.
The first is standard and follows in the limit as $\epsilon$
goes to zero, by an application of dominated
convergence theorem after multiplying the first equation in
\eqref{e2.3} by a test function and integrating by parts.
To show that $\rho$ satisfies the
second equation we show that
\begin{equation}
\mu = R_t + \overline u R_x=0
\label{e2.15}
\end{equation}
in the sense of measures. This is in
LeFloch \cite{le1} and for completeness we give the details of
his arguments for our special case.
Let $(x,t)\in S_c$ and $u=\frac{x-y(x,t)}{t}$, since $u$
satisfies \eqref{e2.1}, we have
\[
-\frac{(x-y(x,t))}{t^2}-\frac{\partial_{t}y(x,t)}{t}+
u(a,t)\frac{(1-\partial_{x}y(x,t))}{t}=0.
\]
It follows that
\[
\partial_{t}y(x,t) + u \partial_{x}y(x,t)=0.
\]
Now
\[
\partial_{t}R(x,t) + u \partial_{x}R(x,t)=(\frac{dv_0}{dx})(y(x,t)
\{\partial_{t}y(x,t) + u\partial_{x}y(x,t)\}
\]
and we get
\[
\partial_{t}R(x,t) + u \partial_{x}R(x,t)=0.
\]
Now we consider a point $(s(t),t) \in S_j$, then
\[
\frac{ds(t)}{dt} =
\frac{u(s(t)+,t))- u(s(t)-,t)}{2}
\]
is the speed of propagation of the discontinuity at this point.
\[
\begin{aligned}
&\mu\{(s(t),t)\}\\
&=-\frac{ds(t)}{dt}(R(s(t)+,t)-R(s(t)-,t))\\
&+\int_0^1 (u(s(t)-,t)
+\alpha (u(s(t)+,t) - u(s(t)-,t))d\alpha (R(s(t)+,t)-R(s(t)-,t))\\
&=[-\frac{ds(t)}{dt} +
\frac{(u(s(t)+,t))+(u(s(t)-,t))}{2}]
(R(s(t)+,t)-R(s(t)-,t))\\
&=0.
\end{aligned}
\]
This proves \eqref{e2.15}. Since $R_{tx}=R_{xt}$, in the sense of
distributions, and $\rho = R_x$, differentiating \eqref{e2.15}
with respect to $x$ gives
\[
\rho_t +(\overline u \rho)_x =0
\]
in the sense of distributions and \eqref{e2.14} follows.
To show that the solution satisfies the initial conditions, first we
observe by Lax's \cite{la1}
argument, $\lim_{t \to 0} u(x,t)=u_0(x)$, a.e. $x$. Now since $y(x,t)-x
= -t u(x,t)$, it follows that $y(x,t)\to x$ as $t\to 0$ a.e.
$x$. So we get $\int_0^{y(x,t)} v_0(z) dz \to \int_0^x v_0(z)dz$
as $t\to 0$ for a.e $x$. $u$ satisfies the entropy condition
$u(x-0,y)\geq u(x+0,t)$ follows from the increasing nature of
$y^{+}(x,t)$ and $y^{-}(x,t)$ as a function of $x$, for each $t>0$ and
the formula \eqref{e2.10} for $u$.
The proof of the theorem is complete.
\end{proof}
It is instructive to deduce the formula for $(u,\rho)$ obtained in
\cite{j2} for special initial data namely the Riemann initial data
from \eqref{e2.10}, \eqref{e2.11} and \eqref{e2.12}. LeFloch \cite{le1}
noted that there are infinite number of solutions for the Riemann
problem, here we select the one given by the vanishing viscosity
solution.
\subsection{Formula for some special initial data}
First we take Riemann type initial data, namely
\begin{equation}
u_0(x)= \begin{cases}
u_l, &\text{if } x<0,\\
u_r, &\text{if } x>0,
\end{cases}\qquad
\rho_0(x) = \begin{cases}
\rho_l, &\text{if }x<0,\\
\rho_r, &\text{if }x>0
\end{cases}
\label{e2.16}
\end{equation}
We have the following formula for the vanishing viscosity limit.
\begin{theorem} \label{thm2.3}
Let $u^\epsilon$ and $\rho^\epsilon$ are solutions given by \eqref{e2.7}
with initial data of Riemann type \eqref{e2.14} and $u(x,t)=
\lim_{\epsilon
\to 0} u^\epsilon(x,t)$ and $\rho(x,t)= \lim_{\epsilon
\to 0} u^\epsilon(x,t)$, the $(u,\rho)$ have the following form.
\\
\textbf{Case 1 $u_l=u_r =u_0$:}
\begin{gather*}
u(x,t) = u_0\\
\rho(x,t) = \begin{cases}
\rho_l,&\text{if }xu_0 t
\end{cases}
\end{gather*}
\textbf{Case 2 $u_lu_rt
\end{cases}
\]
and
\[
\rho(x,t) = \begin{cases}
\rho_l,&\text{if }xu_r t
\end{cases}
\]
\textbf{Case 3 $u_rst
\end{cases}
\\
d\rho(x,t) = \begin{cases}
\rho_l dx,&\text{if }xst
\end{cases}
\end{gather*}
where $s=\frac{u_l+u_r}{2}$
\end{theorem}
\begin{proof}
By the previous theorem it is sufficient to compute $y(x,t)$ for the
minimizer in \eqref{e2.9}. An easy computation shows that $y(x,t)$
takes the following form for each of the cases:
For case 1, $u_l=u_r =u_0$,
$y(x,t) = x- t u_0$.
For case 2, $u_lu_r t
\end{cases}
\]
and for case $u_rst
\end{cases}
\]
Substituting these values of $y(x,t)$, in \eqref{e2.10} - \eqref{e2.12},
the formula for $(u,\rho)$ follows.
An independent proof that $(u,\rho)$ satisfies the equation is
instructive and is taken from Joseph \cite{j2}. The first equation
is standard, we only deal with the second equation of \eqref{e2.1}. An
easy calculation shows that that
\[
\begin{aligned}
(\rho,\phi_t)+(u\rho, \phi_x)
&=\int_0^\infty (u_l-u_r)(\frac{\rho_l+\rho_r}{2}
t \phi_t(st,t) dt\\
&\quad+\int_0^\infty (\frac{u_l+u_r}{2})(u_l-u_r)(\frac{\rho_l+\rho_r}{2})
t \phi_x(st,t)dt\\
&\quad +\int_{xst}(\rho_r\phi_t +\rho_r u_r \phi_x)dx dt\\
&= \int_0^\infty\{R(u_l,u_r,\rho_l,\rho_r)\}\phi(st,t) dt
\end{aligned}
\]
where
\[
R(u_l,u_r,\rho_l,\rho_r) = (u_l -u_r)(\frac{\rho_l+\rho_r}{2})-
\frac{(\rho_r-\rho_l)(u_l+u_r)-2\rho_r u_r +2\rho_l u_l}{2}
\]
Here we used
\begin{gather*}
\int_{xst}(\rho_r\phi_t +\rho_r u_r \phi_x)dx dt
=\int_0^\infty(s\rho_r-\rho_ru_r)\phi(st,t) dt,
\\
\frac{\phi(st,t)}{dt} = s\phi_x(st,t)+\phi_t(st,t),
\\
\int_0^\infty t \frac{d\phi(st,t)}{dt} dt
=-\int_0^\infty\phi(st,t)dt\,.
\end{gather*}
It is easy to see $R(u_l,u_r,\rho_l,\rho_r)=0$.
\end{proof}
\subsection*{An example with finite mass}
Another interesting case is initial data of finite
mass and of the form
\begin{gather*}
u_0(x) = \begin{cases}
0,&\text{if $-\infty t/2+1
\end{cases}
\end{gather*}
and for $t> 2$:
\begin{gather*}
u(x,t) = \begin{cases}
0,&\text{if }x<0,\\
x/t ,&\text{if }0\leq x \leq (2t)^{1/2}\\
0 ,&\text{if }x \geq (2t)^{1/2}
\end{cases}
\\
d\rho(x,t) =\rho_c \delta_{x=(2t)^{1/2}}
\end{gather*}
Here we remark that the vanishing viscosity limit
\eqref{e2.12} for $\rho$ is an extension of the formula \eqref{e1.5},
or equivalently $\rho(x,t)=\rho_0(y(x,t))
\det(\frac{\partial y(x,t)}{\partial x})$. In \eqref{e1.5} solution
exists only short time where as here the solution is global, $y(x,t)$
is defined through a minimization problem
\eqref{e2.9} and the product has to be interpreted suitably
as a measure. Both are same up to the time smooth solution exists.
\subsection{Generalized solutions in the sense of Colombeau}
We have seen that when initial data $(u_0,\rho_0)$ in the space of
bounded measurable functions which is locally BV $u$
contains a classical shock but
density is not a function; it contain a $\delta$ measure concentrated
along the shock.
As we know $D'$ the space of distributions is not an algebra
there is a problem in the product $\rho u$ with appear in the second
equation. We defined the product as
a Radon measure proposed by Volpert \cite{v1} and used some ideas of
LeFloch \cite{le1} to show it is the solution of the problem
\eqref{e2.1} and \eqref{e2.2} when the initial data is in the
space of bounded measurable functions which is locally a BV function.
In this section we consider a larger class of function as initial data
and adopt the approach of Colombeau \cite{bi1,co1,co2,co3}, and
construct solution for of \eqref{e2.1} and \eqref{e2.2} with equality
replaced by association in the sense of Colombeau. This approach takes
into account not only the final limit but the microscopic structure of
the shock due to the viscous effects in the solutions.
First we describe the algebra of generalized functions of Colombeau in
$\Omega = \{(x,t), x\in R^1, t>0\}$, denoted
by $\mathcal{G}(\Omega)$. Let $C^\infty(\Omega)$, the class of
infinitely differentiable functions
in $\Omega$ and consider the infinite product $\mathcal{E}(\Omega)=
[C^\infty(\Omega)]^{(0,1)}$.
Thus any element $v$ of $\mathcal{E}{(\Omega)}$ is a map from $(0,1)$
to
$C^\infty(\Omega)$
and is denoted by $v=(v^\epsilon)_{0<\epsilon<1}$. An element
$v=(v^\epsilon)_{0<\epsilon<1}$ is called moderate if given a compact
subset K of $\Omega$
and $j$ and $\ell$ non negative integers,there exists $N>0$ such that
\begin{equation}
\| \partial^j_t \partial^{\ell}_x v^\epsilon
\|_{L^\infty (K)} =\mathcal{{O}}(\epsilon^{-N}) ,
\label{e2.17}
\end{equation}
as $\epsilon$ tends to $0$. An element $v=(v^\epsilon)_{0<\epsilon<1}$
is called null
if for all compact subsets K of $\Omega$ and for all nonnegative integers
$j$ and $\ell$ and for all $M>0$
\begin{equation}
\| \partial^j_t \partial^{\ell}_x v^\epsilon \|_
{L^\infty(K)} = \mathcal{{O}}(\epsilon^{M}) ,
\label{e2.18}
\end{equation}
as $\epsilon$ approaches $0$. The set of all moderate elements is denoted
by $\mathcal{E}_M (\Omega)$ and the set of null elements is denoted by
$\mathcal{N}(\Omega)$. It is easy to
see that $\mathcal{E}_M (\Omega)$ is an algebra with partial
derivatives, the
operations being defined point wise on representatives and
$\mathcal{N}(\Omega)$
is an ideal which is closed under differentiation. The quotient space
denoted
by
\begin{equation}
\mathcal{G}{(\Omega)} =\frac{\mathcal{E}_M (\Omega)}{\mathcal{N}(\Omega)}
\label{e2.19}
\end{equation}
is an algebra with partial derivatives, the operations being defined on
representatives. The algebra $\mathcal{G}(\Omega)$ is called the
algebra of generalized functions of Colombeau.
Two elements $u$ and $v$ in $\mathcal{G}(\Omega)$
are said to be associated, if for some (and hence all) representatives
$(u^\epsilon)_{0<\epsilon<1}$ and $(v^\epsilon)_{0<\epsilon<1}$, of $u$
and
$v$ , $u_\epsilon -v_\epsilon$ goes to $0$ as $\epsilon$ tends to $0$,
in
the
sense of distribution and is denoted by "$u \approx v$". Here we remark
that
this notion is different from the notion of equality in $\mathcal{
G}(\Omega)$, which means that $u-v \in \mathcal{N}(\Omega)$, or in
other words,
\[
\| \partial^j_t \partial^{\ell}_x (u^\epsilon-v^\epsilon)
\|_{L^\infty(K)} = \mathcal{{O}}(\epsilon^{M})
\]
for all M, for all compact
subsets K of $\Omega$ for all $j$, and $\ell$ nonnegative integers.\\
We refer to the works \cite{bi1,co1,co2,co3,j5} and the references
therein
that use the Colombeau algebra to find global solutions
of initial value problems when nonconservative product appears
in the equation.
Now, roughly speaking, we show that
$(u,\rho)=(u^\epsilon(x,t))_{0<\epsilon <1},
\rho=(\rho^\epsilon(x,t))_{0<\epsilon <1}$,
with $u^\epsilon$ and $\rho^\epsilon$ given by \eqref{e2.7} satisfies
the equation \eqref{e2.1}, in the quarter plane $\{(x,t) : x \in R^1,
t>0 \}$
in the sense of association:
\begin{equation}
\begin{gathered}
u_t + (\frac{u^2}{2})_x \approx 0\\
v_t + (u\rho)_x \approx 0.
\end{gathered} \label{e2.20}
\end{equation}
More precisely, consider $(u,v)$ where
$u=(u^\epsilon(x,t))_{0<\epsilon <1}$,
and
$\rho=(\rho^\epsilon(x,t))_{0<\epsilon <1}$,
with $u^\epsilon$ and $\rho^\epsilon$ are solutions of
equation \eqref{e2.3}
\begin{equation}
\begin{gathered}
u^\epsilon_t + (\frac{{u^\epsilon}^2}{2})_x =\frac{\epsilon}{2}
u^\epsilon_{xx},\\
\rho^\epsilon_t + (u^\epsilon \rho^\epsilon)_x =\frac{\epsilon}{2}
\rho^\epsilon_{xx},
\end{gathered}
\label{e2.21}
\end{equation}
in $\{(x,t) : x \in R^1, t>0 \}$, supplemented with an
initial
condition at $t=0$
\begin{equation}
u^\epsilon(x,0) = u^\epsilon_{0}(x) , \quad
\rho^\epsilon(x,0) = \rho^\epsilon_{0}(x), \label{e2.22}
\end{equation}
where
$u_0=(u_0^\epsilon(x))_{0<\epsilon <1}$,
$\rho_0=(v_0^\epsilon(x))_{0<\epsilon <1}$,
are in $\mathcal{G}(R^1)$, the algebra
of generalized functions of Colombeau. We assume that $u_0^\epsilon$,
and $v_0^\epsilon$ are bounded $C^\infty$ functions of $x$
with the following estimates, for $j=0,1,2,\dots$,
\begin{equation}
\|{\partial_x}^j u_0^\epsilon\|_{L^\infty([0,\infty))}
= O(\epsilon^{-j})
\|{\partial_x}^j \rho_0^\epsilon\|_{L^\infty([0, \infty))}=
O(\epsilon^{-j})
\label{e2.23}
\end{equation}
and $u_0^\epsilon(x) \to u_0(x)$,
$\rho_0^\epsilon(x) \to \rho_0(x)$
point wise a.e.
These conditions are satisfied for example if we take bounded
measurable functions on $R^1$, and then take convolution with
the Friedrichs mollifiers with scale $\epsilon$.
We shall prove the following result.
\begin{theorem}
Assume that $u_0= (u_0^\epsilon(x)_{0<\epsilon <1}$,
$\rho_0=(\rho_0^\epsilon(x)_{0<\epsilon <1}$, are in
$\mathcal{G}(R^1)$,
with the estimates \eqref{e2.23} and as described before. Let
$(u^\epsilon, \rho^\epsilon)$ be given by the formula \eqref{e2.7}
with $(u_0(x),\rho_0(x))$ replaced by
$(u_0^\epsilon(x),\rho_0^\epsilon(x))$, for $\epsilon>0$,
then $ u= (u^\epsilon)_{0<\epsilon <1}$ and
$ \rho= (\rho^\epsilon)_{0<\epsilon <1}$ are in $\mathcal{G}(\Omega)$
and $(u,v)$ is a solution to \eqref{e2.20} with initial condition
$(u_0,\rho_0)=(u_0^\epsilon(x),\rho_0^\epsilon(x)_{0<\epsilon <1}$
\end{theorem}
\begin{proof}
To show that $u$ and $\rho$ are in $\mathcal{G}(\Omega)$
we need to prove that $u^\epsilon$ and $\rho^\epsilon$
satisfies the estimate \eqref{e2.17}.
From the formulas \eqref{e2.7} and
and the estimate \eqref{e2.23}, it is easy to see the estimates
\begin{equation}
\| u^\epsilon\|_{L^{\infty}(\Omega)} \leq
\| u_0 \|_{L^{\infty}(R^N)} , \quad
\| \rho^\epsilon\|_{L^{\infty}(\Omega)} \leq
\| \rho_0 \|_{L^{\infty}(R^N)}.
\label{2.24}
\end{equation}
An application of the Leibinitz's rule and the estimate \eqref{e2.23},
gives us
\begin{equation}
\| \partial^{k}_{x} u^{\epsilon} \|_
{L^{\infty}(\Omega)} = \mathcal{O}(\epsilon^{-2 k}) , \quad
\| \partial^{\alpha}_{x} \rho^{\epsilon} \|_
{L^{\infty}(\Omega)} = \mathcal{O}(\epsilon^{-2 k}).
\label{e2.25}
\end{equation}
Since $(u^\epsilon,\rho^\epsilon)$ is a solution
of \eqref{e2.21}, using the estimate \eqref{e2.25}, we get
\begin{equation}
\|\partial_tu^\epsilon\|_{L^{\infty}(\Omega)} =
\mathcal{O}(\epsilon^{-2}), \quad
\|\partial_t \rho^\epsilon\|_{L^{\infty}(\Omega)} =
\mathcal{O}(\epsilon^{-2}).
\label{e2.26}
\end{equation}
Now applying the differential operator ${\partial_t} ^{j} {\partial_x} ^
{k}$ on both sides of \eqref{e2.21}, first $k=1,j=0,1,2,
\dots $ and
then $k=2$, $j=0,1,2, \dots $ successively and using \eqref{e2.25}
and \eqref{e2.26} we obtain
\begin{gather*}
\| \partial^{j}_{t} \partial^{k}_{x} u^{\epsilon}
\|_{L^{\infty}(\Omega)} = \mathcal{O}(\epsilon^{-2(j+k)})
\\
\| \partial^{j}_{t} \partial^{k}_{x} \rho^{\epsilon} \|_
{L^{\infty}(\Omega)} = \mathcal{O}(\epsilon^{-2(j+k)})
\end{gather*}
These estimates show that $u$ and $v$ are in $\mathcal{G}(\Omega)$.
Now to show that $u$ and $v$ satisfy \eqref{e2.20} in the
sense of association. We multiply \eqref{e2.21} by a test
function $\phi \in C_0^\infty(\Omega)$ and integrate we get
\begin{gather*}
\int_{0}^{\infty}\!\!\int_{-\infty}^\infty(u^\epsilon_t+
(1/2)({u^\epsilon}^2)_x
\phi\,dx\,dt=
\frac{\epsilon}{2}\int_{0}^{\infty}\!\!\int_{-\infty}^\infty \
u^\epsilon
\phi_{xx} \,dx\,dt,
\\
\int_{0}^{\infty}\!\!\int_{-\infty}^\infty(\rho^\epsilon_t+ (u^\epsilon
\rho^\epsilon)_{x}\phi \,dx\,dt
=-\frac{\epsilon}{2}
\int_{0}^{\infty}\!\!\int_{-\infty}^\infty \frac{b^\epsilon}{a^\epsilon}
{\phi}_{xxx}\,dx\,dt.
\end{gather*}
Now we have to show that the right hand side approaches
zero as $\epsilon$ approaches zero. This easily follows by an
application of dominated convergence theorem
as $u^\epsilon(x,t)$ and $\epsilon {b^\epsilon}/{a^\epsilon}$ are
bounded and
converge point wise almost every where. This completes the proof of
the theorem.
\end{proof}
\section{Boundary value problem}
In this section we consider the system
\begin{equation}
\begin{gathered}
u_t+(u^2/2)_x=0,\\
\rho_t +(\rho u)_x =0
\end{gathered} \label{e3.1}
\end{equation}
in the quarter plane
$\{(x,t) : x >0, t>0 \}$, supplemented with an
initial condition at $t=0$
\begin{equation}
\begin{pmatrix}
u(x,0)\\
\rho(x,0)
\end{pmatrix}
= \begin{pmatrix}
u_0(x)\\
\rho_0(x)
\end{pmatrix}
\label{e3.2}
\end{equation}
and a weak form of the Dirichlet boundary condition,
\begin{equation}
\begin{pmatrix}
u(0,t)\\
\int_0^\infty \rho(y,t) dy
\end{pmatrix}
= \begin{pmatrix}
u_b(t)\\
R_b(t)
\end{pmatrix}
\label{e3.3}
\end{equation}
where $u_0(x)$ and $\rho_0(x)$ are integrable functions of $x$
and
$u_b(t)$ and $R_b(t)$ are Lipschitz continuous functions of $t$. Indeed
with strong form of Dirichlet boundary conditions \eqref{e3.3}, there is
neither existence nor uniqueness as the speed of propagation
$\lambda =u$ and does not have
a definite sign at the boundary $x=0$. We note that the speed is
completely determined by the first equation. We use the Bardos Leroux and
Nedelec
\cite{b1} formulation of the boundary condition for the $u$ component
which for our case is
equivalent to the following condition :
\begin{equation}
u(0+,t) \in E(u_B(t))
\label{e3.4}
\end{equation}
where the admissible set $E(u_B(t))$ is defined by
\[
E(u_B(t))= \begin{cases}
(-\infty,0],&\text{if }u_B(t)\leq 0,\\
(-\infty, -u_B(t))\cup \{u_B(t)\},&\text{if }u_B(t)>0
\end{cases}
\]
Here
$u_{b}^{+}(t)= \max{u_b(t),0}$. There are explicit
representations of the entropy weak solution of of the
first component $u$ of \eqref{e1.1} with initial condition
$u(x,0)=u_0(x)$ and the
boundary condition \eqref{e3.4} by Joseph \cite{j1} and
by Joseph and Gowda \cite{j3}.
We use the formula in \cite{j3} for $u$ which involve a
minimization of functionals on certain class of paths
and generalized characteristics. Once $u$ is
obtained, the equation for $v$ is linear equation with a discontinuous
coefficient $u(x,t)$. Now $\int_0^\infty\rho(0+,t)dt =R_b(t)$ is
prescribed only if
the characteristics at $(0,t)$ has positive speed, i.e., $u(0+,t)>0$.
So the weak form of boundary conditions for $\rho$ component is
\begin{equation}
\text{if $u(0+,t)>0$, then $\int_0^\infty \rho(y,t)dy =R_b(t)$}.
\label{e3.5}
\end{equation}
To state the results we introduce some notation.
For each fixed $(x,y,t)$, $x> 0$, $y \geq 0$, $t>0$,
$C(x,y,t)$ denotes the following class of paths $\beta$ in
the quarter plane
$D=\{ (z,s) : z\geq 0, s \geq 0\}$. Each path is connected from the
initial point
$(y,0)$ to $(x,t)$ and is of the form $z=\beta(s)$, where $\beta$ is a
piecewise linear function of maximum three lines and always linear in
the interior of $D$. Thus for $x>0$ and $y>0$, the curves are
either a straight line or have exactly three straight lines with one
lying on the boundary $x=0$. For $y=0$ the curves are made up
of one straight line or two straight
lines with one piece lying on the
boundary $x=0$. Associated with the data $u_b(t)$, we define the
functional $J(b)$
on $C(x,y,t)$
\begin{equation}
J(\beta) = -\int_{\{s:\beta(s)=0\}}\frac{(u_B(s)^{+})^2}{2}ds +
\int_{\{s:\beta(s)
\neq 0\}}\frac{(\frac{d\beta(s)}{ds})^2}{2}ds.
\label{e3.6}
\end{equation}
We call $\beta_0$ is straight line path connecting $(y,0)$ and $(x,t)$
which does not touch the boundary $x=0$, $\{(0,t), t>0\}$, then let
\begin{equation}
A(x,y,t)= J(\beta_0) = \frac{(x-y)^2}{2t}.
\label{e3.7}
\end{equation}
For any $\beta \in C^{*}(x,y,t) = C(x,y,t)-{\beta_0}$, that
is made up of three straight lines
connecting $(y,0)$ to $(0,t_1)$ in the interior and $(0,t_1)$ to
$(0,t_2)$ on the boundary and $(0,t_2)$ to $(x,t)$ in the interior,
it can be easily seen from \eqref{e3.6} that
\begin{equation}
J(\beta) = J(x,y,t,t_1,t_2) =
-\int_{t_1}^{t_2}\frac{(u_B(s)^{+})^2}{2}ds +
\frac{y^2}{2 t_1} + \frac{x^2}{2(t-t_2)^2}.
\label{e3.8}
\end{equation}
For the curves made up two straight
lines with one piece lying on the
boundary $x=0$ which connects $(0,0)$ and $(0,t_2)$ and the other
connecting $(0,t_2)$ to $(x,t)$.
\[
J(\beta) = J(x,y,t,t_1=0,t_2) =
-\int_{0}^{t_2}\frac{(u_B(s)^{+})^2}{2}ds +
\frac{x^2}{2(t-t_2)^2}.
\]
It was proved in \cite{j1,j3}, that there exists a
$\beta^{*} \in C^{*}(x,y,t)$ or correspondingly $t_1(x,y,t)$,
$t_2(x,y,t)$ so that
\begin{equation}
\begin{aligned}
B(x,y,t)& =J(\beta^{*})\\
&=\min \{J(\beta) :\beta \in C^{*}(x,y,t)\}\\
&= \min \{J(x,y,t,t_1,t_2): \,\, 0\leq t_1 < t_2 < t\}\\
&= J(x,y,t,t_1(x,y,t),t_2(x,y,t))
\end{aligned}
\label{e3.9}
\end{equation}
is a Lipschitz continuous so that
\begin{equation}
Q(x,y,t)= \min\{J(\beta) : \beta \in C(x,y,t)\}
= \min \{A(x,y,t),B(x,y,t)\},
\label{e3.10}
\end{equation}
and
\begin{equation}
U(x,t)= \min \{Q(x,y,t) + U_0(z), \,\, 0\leq y< \infty\}
\label{e3.11}
\end{equation}
are Lipschitz
continuous functions in their variables,
where $U_0(y)=\int_0^y u_0(z)dz$.
Further minimum in \eqref{e3.11} is attained at some value 0f $y\geq 0$
which
depends on $(x,t)$, we call it $y(x,t)$. If
$A(x,y(x,t),t)\leq B(x,y(x,t),t)$
\begin{equation}
U(x,t)= \frac{(x-y(x,t))^2}{2t}) + U_0(y),
\label{e3.12}
\end{equation}
and if $A(x,y(x,t),t)>B(x,y(x,t),t)$,
\begin{equation}
U(x,t)=J(x,y(x,t),t,t_1(x,y(x,t),t),t_2(x,y(x,t),t))
+ U_0(y).
\label{e3.13}
\end{equation}
Here and hence forth $y(x,t)$ is a
minimizer in \eqref{e3.11} and in the case of
\eqref{e3.13}, $t_2(x,t)=t_2(x,y(x,t),t)$ and
$t_1(x,t)=t_1(x,y(x,t),t)$.
\begin{theorem}\label{thm3.1}
For every $(x,t)$ minimum in \eqref{e3.11}
is achieved by some $y(x,t)$, and $U(x,t)$ is a
Lipschitz continuous and for almost every $(x,t)$ there is only one
minimizer $y(x,t)$.
For every point $(x,t)$ satisfying
$U(x,t)=A(x,y(x,t),t)\leq B(x,y(x,t),t)$, define
\begin{gather*}
u(x,t)= \frac{x-y(x,t)}{t}\\
\rho(x,t)= \partial_x(R_0(y(x,t)).
\end{gather*}
and for the points $(x,t)$ where $B(x,y(x,t),t)0$.
The explicit solution shows that the order of singularities of solutions
of \eqref{e4.1} increases as $n$ increases in the passage to the limit
$\epsilon\to 0$.
When $n=1$, and $u=u_1$, \eqref{e4.1} becomes the Burgers
equation \eqref{e1.1} and was studied by Hopf \cite{h1}
as $\epsilon\to 0$. The limit
is bounded $BV_{\rm loc}$ functions if initial data is bounded and is
a solution to the inviscid Burgers equation which is an example of
conservation laws with genuinely nonlinear characteristic fields.
Here we have classical shocks.
When $n=2$ and $u= u_1$, $\rho =u_2$, \eqref{e4.1} is the modified
adhesion model \eqref{e2.3}, where
$u$ remains bounded but $\rho$ is not bounded functions but a
Radon measure, in the
limit $\epsilon \to 0$, when the initial data
are bounded measurable functions which are locally BV.
When $n=3$ and $u= u_1$, $\rho =u_2$, $w=u_3$, in \eqref{e4.1} the
equation becomes
\begin{gather*}
u_t+(u^2/2)_x=\frac{\epsilon}{2} u_{xx},\\
\rho_t +(\rho u)_x=\frac{\epsilon}{2}\rho_{xx}\\
w_t +(\rho^2 + u w)_x=\frac{\epsilon}{2} w_{xx}.
\end{gather*}
Joseph \cite {j5} observed that the nature of singularity is
worse than $\delta$ -measures
and constructed solution in the algebra of
generalized functions of Colombeau. Shelkovich \cite{sh1} used
the explicit solution of \cite{j5} to derive
a formula for the solution for the inviscid case,
in the limit $\epsilon \to 0$
and showed that the solution contains
$\delta$ and its derivative
$\delta'$ when the initial data is of Riemann type.
The exact nature of singularities of solutions of \eqref{e4.1}, as
$\epsilon$ goes to zero
is still not known for the cases $n\geq 4$, even with Riemann type
initial data.
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\section*{Addendum posted on September 20, 2013}
I would like to correct a mistake in the statement of Theorem \ref{thm3.1}
on page 13; the proof remains the same.
\subsection*{New Theorem 3.1} {\it
For every point $(x,t)$ in the quarter plane $\{(x,t):x>0,t>0 \}$ the minimum in
\eqref{e3.11} is achieved by some $y(x,t)$, and $U(x,t)$ is a
Lipschitz continuous. Further the minimizer $y(x,t)$ is unique for almost
every $(x,t)$. With $A(x,y,t)$, $B(x,y,t)$ given by \eqref{e3.7} and \eqref{e3.9},
define
\begin{gather*}
u(x,t) = \begin{cases}
\frac{x-y(x,t)}{t}, &\text{if }A(x,y(x,t),t)\leq B(x,y(x,t),t) ,\\[3pt]
\frac{x}{t-t_2(x,t)}, &\text{if }B(x,y(x,t),t)