\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 52, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2011/52\hfil Solutions for a system of PDEs]
{Explicit solutions for a system of first-order partial differential
equations-II}
\author[K. T. Joseph, M. R. Sahoo\hfil EJDE-2011/52\hfilneg]
{Kayyunnapapra Thomas Joseph, Manas Ranjan Sahoo} % in alphabetical order
\address{Kayyunnapapra Thomas Joseph \newline
School of Mathematics\\
Tata Institute of Fundamental Research\\
Homi Bhabha Road\\
Mumbai 400005, India}
\email{ktj@math.tifr.res.in}
\address{Manas Ranjan Sahoo \newline
School of Mathematics\\
Tata Institute of Fundamental Research\\
Homi Bhabha Road\\
Mumbai 400005, India}
\email{manas@math.tifr.res.in}
\thanks{Submitted April 8, 2011. Published April 13, 2011.}
\subjclass[2000]{35A20, 35L50, 35R05}
\keywords{First order PDE; boundary conditions; exact solutions}
\begin{abstract}
In this note we give an explicit formula for the solution of
conservative form of a system studied in a previous article \cite{j5},
in the domain $\{(x,t):x>0,t>0\}$ with initial conditions at
$t=0$ and with Bardos Leroux Nedelec boundary conditions at $x=0$.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\section{Introduction}
In this note we consider the conservative form of a system
considered in \cite{j5}, namely
\begin{equation}
\begin{gathered}
u_t + f(u)_x =0,\\
v_t + (f'(u) v)_x =0,
\end{gathered}
\label{e1.1}
\end{equation}
with $f''(u)>0$, in the domain $\Omega = \{(x,t) : x>0, t>0 \}$.
We give an explicit formula for the solution of \eqref{e1.1}
with prescribed initial conditions
\begin{equation}
\begin{pmatrix}
u(x,0)\\
v(x,0)
\end{pmatrix}
= \begin{pmatrix}
u_0(x)\\
v_0(x)
\end{pmatrix},
\label{e1.2}
\end{equation}
at $t=0$, the Bardos Leroux and Nedelec
\cite{b1,le2} boundary condition for $u$
\begin{equation}
\begin{gathered}
\text{either}\quad u(0+,t)= u_{b}^{+}(t)\\
\text{or}\quad f'(u(0+,t))\leq 0 \text{ and }
f(u(0+,t))\geq f(u_{b}^{+}(t)),
\end{gathered}\label{e1.3}
\end{equation}
and a weak form of Dirichlet boundary conditions for $v$
\begin{equation}
\text{if $f'(u(0+,t))>0$, then $v(0+,t)=v_b(t)$}.
\label{e1.4}
\end{equation}
Here
$u_{b}^{+}(t)= \max\{u_b(t),\lambda\}$, where
$\lambda$ is the unique point where $f'(u)$ changes sign.
In \cite{j5}, explicit solution was constructed for the system
where the second equation in \eqref{e1.1} was replaced by
\begin{equation}
V_t +f'(u)V_x =0
\label{e1.5}
\end{equation}
with the weak form of Dirichlet boundary condition $V(0,t)=V_b(t)$.
Taking
derivative of \eqref{e1.5} withe respect to $x$ and setting
$v=V_x$ we obtain the conservative equation for $v$.
In this note we give Dirichlet boundary condition for $v$,
which is equivalent to giving
Neumann Boundary condition for $V$. Here we explain the required
modification of the formula in \cite{j5} in the construction of
solution to \eqref{e1.1}-\eqref{e1.4}.
LeFloch \cite{le1} was the first who studied the system \eqref{e1.1}
when $f(u)$ is strictly convex and constructed
explicit formula for the pure initial value problem using Lax formula.
One important property of the system is the formation
of $\delta$ - wave solutions for certain types of initial data which
are of bounded variation. Such systems come in applications, for example,
the special case $f(u)=u^2/2$ in \eqref{e1.1}, is the one-dimensional
model in the large scale structure formation. Initial value problem for
this quadratic case was
also studied by Joseph \cite {j1,j4} by different way, using the
vanishing viscosity method
and Hopf-Cole transformation.
\section{A formula for the solution in the quarter
plane}
We consider the system \eqref{e1.1}
with initial condition \eqref{e1.2} and
boundary condition \eqref{e1.3} and \eqref{e1.4}. We assume
$u_0(x)$ is bounded measurable and $v_0(x)$ is Lipschitz
continuous functions of $x \geq 0$,
$u_b(t)$ and $v_b(t)$ are Lipschitz continuous functions of $t>0$.
We assume the flux $f(u)$ satisfies the conditions
\[
f''(u)>0, \quad \lim_{u \to \infty}\frac{f(u)}{u} = \infty,
\]
and let $f^{*}(u)$ be the convex dual of $f(u)$ namely,
$f^{*}(u)= \max_{\theta\in R^1}\{\theta u -f(\theta)\}$.
As in \cite{j5}, we introduce some notation and describe the
construction of $(u,v)$ and then verify it
is a solution. For each fixed $(x,y,t), x \geq 0, y \geq 0, t>0$,
$C(x,y,t)$ denotes the following class of paths $\beta$ in
the quarter plane
$\Omega=\{ (z,s) : z\geq 0, s \geq 0\}$.
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 $\Omega$. 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
flux $f(u)$ and boundary data $u_b(t)$, we define the functional
$J(\beta)$
on $C(x,y,t)$
\[
J(\beta) = -\int_{\{s:\beta(s)=0\}}f(u_B(s)^{+})ds + \int_{\{s:\beta(s)
\neq 0\}}f^{*}\big(\frac{d\beta(s)}{ds}\big)ds.
\]
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
\[
A(x,y,t)= J(\beta_0) =t f^* \big(\frac{x-y}{t}\big).
\]
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 $(0,t_2)$ in the interior and $(0,t_2)$ to
$(0,t_1)$ on the boundary and $(0,t_1)$ to $(x,t)$ in the
interior, $t_20$, this
minimizer is unique except for a countable number of points of $x>0$.
Finally, for each fixed $t>0$, except for one point of $x$, either
$A(x,y(x,t),t)< B(x,y(x,t),t)$ or $A(x,y(x,t),t)> B(x,y(x,t),t)$.If
$A(x,y(x,t),t)< B(x,y(x,t),t)$,
\[
U(x,t)=tf^{*}(\frac{x-y(x,t)}{t}) + U_0(y),
\]
and if $A(x,y(x,t),t)>B(x,y(x,t),t)$
\[
U(x,t)=J(x,y(x,t),t,t_1(x,y(x,t),t),t_2(x,y(x,t),t))
+ U_0(y).
\]
Here and hence forth $y(x,t)$ is a
minimizer in \eqref{e2.1} and we denote
$A(x,t)=A(x,y(x,t),t)$,
$B(x,t)=B(x,y(x,t),t)$,$t_2(x,t)=t_2(x,y(x,t),t)$
and
$t_1(x,t)=t_1(x,y(x,t),t)$.
\begin{theorem}\label{thm2.1}
Assume $u_0$ is bounded measurable and locally Lipschitz continuous,
$v_0$ is Lipschitz continuous in $x \geq 0$ and $u_b(t)$ ans $v_b(t)$
are Lipschitz continuous functions. Then
for every $\{(x,t),x \geq 0, t>0$, $U(x,t)$ defined by the minimization
problem \eqref{e2.1} is a locally
Lipschitz continuous function. For almost every $(x,t)$ there is
only one minimizer $y(x,t)$ and let $t_1(x,t)$ and $t_2(x,t)$ as
described before. Define
\begin{equation}
u(x,t) = \begin{cases}
(f^{*})'(\frac{x-y(x,t)}{t}),&\text{if } A(x,t)< B(x,t),\\
(f^{*})'(\frac{x}{t-t_1(x,t)}),&\text{if } A(x,t)> B(x,t),
\end{cases}
\label{e2.2}
\end{equation}
and
\begin{equation}
V(x,t) = \begin{cases}
\int_0^{y(x,t)} v_0(z) dz ,&\text{if } A(x,t)< B(x,t),\\
-\int_{t_2(x,t)}^{t_1(x,t)} f'(u_{b}^{+}(s) v_b(s) ds,&\text{if }
A(x,t)> B(x,t),
\end{cases}
\label{e2.3}
\end{equation}
and set
\begin{equation}
\begin{gathered}
v(x,t)= \partial_{x}(V(x,t)).
\end{gathered}
\label{e2.4}
\end{equation}
Then the function $(u(x,t),v(x,t))$ is a weak solution of \eqref{e1.1},
satisfying the initial condition \eqref{e1.2} and boundary conditions
\eqref{e1.3} and \eqref{e1.4}. Further $u$ satisfies the entropy
condition $u(x-,t) \geq u(x+,t)$ for $x>0$, $t>0$.
\end{theorem}
\begin{proof}
The proof is by direct verification and most part is identical to
\cite{j5} and so that part is omitted. We give here only the
verification of
the boundary condition \eqref{e1.4}.
Suppose
$f'(u(0+,t))>0$ then $f'(u(x,t))>0$ for $00.
\label{e2.5}
\end{equation}
Now
\begin{equation}
\begin{split}
v(x,t) &= -\partial_{x}\int _{t_2(x,t)}^{t_1(x,t)} f'(u_b^{+}(s)
v_b(s) ds.\\
&= -f'(u_b(t_1 (x,t)))v_b (t_1 (x,t)) \partial_{x} t_1 (x,t)
\end{split} \label{e2.6}
\end{equation}
Again differentiating the relation $t-t_1(x,t) = x/f'(u(x,t))$
with respect to $x$, we have
\begin{equation}
\partial_{x}t_1
(x,t)=\frac{xf''(u(x,t))u_x-f'(u(x,t))}{(f'(u(x,t)))^2}
\label{e2.7}
\end{equation}
By \eqref{e2.5}-\eqref{e2.7} and using the fact
$\lim_{x\to 0}t_1(x,t)=t$, we get the weak boundary condition
\eqref{e1.4}.
\end{proof}
\subsection*{Explicit formula for Riemann initial boundary
value problem}
It is illustrative to compute the solution constructed in the above
theorem for the Riemann type initial boundary data, namely $u_0$, $v_0$,
$u_b$ and $v_b$ are all constants.
\begin{theorem} \label{thm2.2}
For Riemann initial boundary value problems, the
formulae \eqref{e2.2} - \eqref{e2.4} takes the form
{Case 1: $f'(u_0)=f'(u_b) >0$,}
\[
(u(x,t),v(x,t)) = \begin{cases}
(u_0,v_b), &\text{if } x < f'(u_0) t,\\
(u_0, v_0),&\text{if } x > f'(u_0) t.
\end{cases}
\]
{Case 2: $f'(u_0)=f'(u_b) <0$,}
\[
(u(x,t),v(x,t) = (u_0,v_0)
\]
{Case 3: $0f'(u_0) t
\end{cases}
\]
{Case 4: $f'(u_b)<0f'(u_0) t
\end{cases}
\]
{Case 5: $f'(u_b)<0$ and $f'(u_0)\leq 0$,}
\[
(u(x,t),v(x,t)) = (u_0,v_0)
\]
{Case 6: $f'(u_0)0$ :}
\[
(u(x,t),v(x,t)) = \begin{cases}
(u_b,v_b),&\text{if }xst.
\end{cases}
\]
\end{theorem}
\section{Solution in a strip}
The solution we have obtained for the quarter plane problem can be
easily generalized to the ship $\Omega =\{(x,t): 00 \}$.
Here we prescribe
\begin{equation}
(u(x,0+),v(x,0+) = (u_0(x),v_0(x)),\quad 0\leq x\leq 1.
\label{e3.1}
\end{equation}
As before for $u$ component we prescribe a weak form of Dirichlet
boundary conditions at $x=0$ and at $x=1$:
\begin{equation}
\begin{gathered}
\text{either}\quad u(0+,t)= u_{l}^{+}(t)\\
\text{or}\quad f'(u(0+,t))\leq 0 \text{ and }
f(u(0+,t))\geq f(u_{b}^{+}(t)),
\end{gathered}\label{e3.2}
\end{equation}
\begin{equation}
\begin{gathered}
\text{either}\quad u(1-,t)= u_{r}^{+}(t)\\
\text{or}\quad f'(u(1-,t))\geq 0 \text{ and }
f(u(1-,t))\geq f(u_{r}^{+}(t)).
\end{gathered}\label{e3.3}
\end{equation}
Here
$u_{l}^{+}(t)= \max\{u_l(t),\lambda\}$,
$u_{r}^{-}(t)= \min\{u_r(t),\lambda\}$ where as before $\lambda$
is the point of minimum of $f$. We get explicit formula for
the entropy weak solution of the
first component $u$ of \eqref{e1.1} with initial condition
$u(x,0)=u_0(x)$ and the
boundary conditions \eqref{e3.2} and \eqref{e3.3} by Joseph and Gowda
\cite{j3}. Once $u$ is obtained, the boundary conditions for $v(0+,t)
=v_l(t)$ is prescribed only if
the characteristics at $(0,t)$ has positive speed, ie $f'(u(0+,t))>0$.
So the weak form of boundary conditions for $v$ component at $x=0$ is
\begin{equation}
\text{if $f'(u(0+,t))>0$, then $v(0+,t)=v_l(t)$.}
\label{e3.4}
\end{equation}
Similarly the weak form of the boundary condition at $x=1$ is
\begin{equation}
\text{if $f'(u(1-,t))<0$, then $v(1-,t)=v_r(t)$.}
\label{e3.5}
\end{equation}
We assume the initial conditions $u_0(x)$ is bounded measurable, and
locally Lipschitz,
and $v_0(x)$ is Lipschitz continuous on $0\leq x\leq 1$ and boundary
datas $u_l(t),v_b(t)$ are Lipschitz continuous $[0,T]$,
for each $T>0$.
For the statement of the theorem, we introduce some notations.
For each fixed $(x,y ,t)$, $0\leq x \leq1$, $0\leq y \leq1$,
$t>0$, $|i-j|\leq 1$,$i,j=0,1,2,3,\dots$,
$C_{ij}(x,y,t)$ denotes the
following class of paths $\beta$ in the strip
\begin{equation*}
\Omega=\{(z,s):0\leq z\leq 1 ,s\geq 0\}
\end{equation*}
Each path connects $(y,0)$ to $(x,t)$ and is of the form $z=\beta(s)$
where $\beta(s)$ is piecewise linear function which are straight
lines in the interior of $D$ , and having $i$ straight line pieces lie
on $x=0$ and $j$ of them lie on $x=1$ . The points of intersection of
the straight line pieces of the curve lying in $\Omega$ with
the boundaries $x=0$ and $x=1$ are called corners of the curve
$\beta$.
Denote
\begin{equation*}
C(x,y,t)=\cup_{i\geq 0,j\geq 0,|i-j|\leq 1}C_{i,j}(x,y,t)
\end{equation*}
For fixed $(x,y,t)$, we define
\begin{equation}
J(\beta)=-\int_{\{s:\beta(s)=0\}}f(u_l^+ (s))ds -\int_{\{s:\beta(s)=1\}}f(u_r^- (s))ds
+\int_{\{s:0<\beta(s)<1\}}f^*(\frac{d\beta}{ds})ds.
\label{e3.6}
\end{equation}
Denote $C^*(x,y,t)=C(x,y,t)-\{\beta_0 \}$, where $\beta_0$ is the
straight
line path joining $(x,t)$ to $(y,0)$.
Let us define $A(x,y,t)$ and $B(x,y,t)$ by
\begin{equation}
A(x,y,t) = J(\beta_0),\quad
B(x,y,t)=\min_{\beta \in C^*(x,y,t)} J(\beta)
\label{e3.7}
\end{equation}
where $J(\beta)$ be defined by \eqref{e3.6}.
We recall a few facts from \cite{j3}. For each $(x,t) \in \Omega$
and $0\leq y\leq 1$, the minimum in \eqref{e3.7}
is attained for a path $\beta^*$ over $C^*(x,y,t)$. Let
the corner points of the minimizer $\beta$
be
\begin{gather*}
(\beta^*(t_1(x,y,t)),t_1(x,y,t)), \quad
(\beta^*(t_2(x,y,t)),t_2(x,y,t)), \\
\dots, \quad
(\beta^*(t_k(x,y,t)),t_k(x,y,t)),
\end{gather*}
$t> t_1(x,y,t)>t_2(x,y,t)\dots>t_k(x,y,t)>0$. For a
given $T>0$, there exits positive integer $N(T)$ such that for
any $t\leq T$,
and $kB(x,y(x,t)t)$ in which case $U(x,t)=\int_0^y
u_0(z)dz+B(x,y,t)$.
In the second case, let $t_j(x,y,t),j=1,2,\dots k$ corresponds to
the corner points of the curve $\beta^*$ in the evaluation of
$B(x,y,t)$. Denote $t_j(x,t)=t_j(x,y(x,t),t)$,
$A(x,t)=A(x,y(x,t),t)$ and $B(x,t)=B(x,y(x,t),t)$.
With these notations we have the following theorem.
\begin{theorem} \label{thm3.1}
Let $U$ be defined by the minimization
problem \eqref{e3.9} and $y(x,t)$ be a minimizer (which is unique for
a.e points of $\Omega$). Let
$u= U_x(x,t)$ exists for a.e. points of $\Omega$ and has the form
\begin{equation*}
u(x,t) = \begin{cases}
(f^*)'(\frac{x-y(x,t)}{t}), & \text{if } A(x,t)< B(x,t),\\
(f^*)'(\frac{x}{t-t_1(x,t)}), & \text{if } A(x,t)> B(x,t),
\end{cases}
\end{equation*}
and
\begin{equation*}
V(x,t) = \begin{cases}
\int_{0}^{y(x,t)}v_0(z) dz, & \text{if } A(x,t)< B(x,t),\\
-\int _{t_2(x,t)}^{t_1(x,t)} f'(u_l^{+}(s) v_l(s) ds,&\text{if }
A(x,t)> B(x,t) \text{ and } \beta^* (t_1(x,t))=0 ,\\
-\int _{t_2(x,t)}^{t_1(x,t)} f'(u_r^{-}(s) v_r(s) ds,
&\text{if } A(x,t)> B(x,t) \text{ and } \beta^* (t_1(x,t))=1,
\end{cases}
\end{equation*}
and set
\begin{equation*}
v(x,t)= \partial_{x}(V(x,t)).
\end{equation*}
Then $(u,v)$ is a solution to \eqref{e1.1} with initial
conditions \eqref{e3.1} and boundary conditions
\eqref{e3.2}-\eqref{e3.5}.
Further $u$ satisfies the entropy
condition $u(x-,t) \geq u(x+,t)$ for $00$.
\end{theorem}
\begin{proof}
The assertions on $u$ is proved in \cite{j3}. Once we have that, the
verification that $v$ solves the equation and the initial and boundary
conditions follows exactly as in section 2 and is omitted.
\end{proof}
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\end{document}