\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 $$\begin{gathered} u_t + f(u)_x =0,\\ v_t + (f'(u) v)_x =0, \end{gathered} \label{e1.1}$$ 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 $$\begin{pmatrix} u(x,0)\\ v(x,0) \end{pmatrix} = \begin{pmatrix} u_0(x)\\ v_0(x) \end{pmatrix}, \label{e1.2}$$ at $t=0$, the Bardos Leroux and Nedelec \cite{b1,le2} boundary condition for $u$ $$\begin{gathered} \text{either}\quad u(0+,t)= u_{b}^{+}(t)\\ \text{or}\quad f'(u(0+,t))\leq 0 \text{ and } f(u(0+,t))\geq f(u_{b}^{+}(t)), \end{gathered}\label{e1.3}$$ and a weak form of Dirichlet boundary conditions for $v$ $$\text{if f'(u(0+,t))>0, then v(0+,t)=v_b(t)}. \label{e1.4}$$ 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 $$V_t +f'(u)V_x =0 \label{e1.5}$$ 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 $$u(x,t) = \begin{cases} (f^{*})'(\frac{x-y(x,t)}{t}),&\text{if } A(x,t)< B(x,t),\\ (f^{*})'(\frac{x}{t-t_1(x,t)}),&\text{if } A(x,t)> B(x,t), \end{cases} \label{e2.2}$$ and $$V(x,t) = \begin{cases} \int_0^{y(x,t)} v_0(z) dz ,&\text{if } A(x,t)< B(x,t),\\ -\int_{t_2(x,t)}^{t_1(x,t)} f'(u_{b}^{+}(s) v_b(s) ds,&\text{if } A(x,t)> B(x,t), \end{cases} \label{e2.3}$$ and set $$\begin{gathered} v(x,t)= \partial_{x}(V(x,t)). \end{gathered} \label{e2.4}$$ 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} Now $$\begin{split} v(x,t) &= -\partial_{x}\int _{t_2(x,t)}^{t_1(x,t)} f'(u_b^{+}(s) v_b(s) ds.\\ &= -f'(u_b(t_1 (x,t)))v_b (t_1 (x,t)) \partial_{x} t_1 (x,t) \end{split} \label{e2.6}$$ Again differentiating the relation$t-t_1(x,t) = x/f'(u(x,t))$with respect to$x$, we have $$\partial_{x}t_1 (x,t)=\frac{xf''(u(x,t))u_x-f'(u(x,t))}{(f'(u(x,t)))^2} \label{e2.7}$$ 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 $$(u(x,0+),v(x,0+) = (u_0(x),v_0(x)),\quad 0\leq x\leq 1. \label{e3.1}$$ As before for$u$component we prescribe a weak form of Dirichlet boundary conditions at$x=0$and at$x=1$: $$\begin{gathered} \text{either}\quad u(0+,t)= u_{l}^{+}(t)\\ \text{or}\quad f'(u(0+,t))\leq 0 \text{ and } f(u(0+,t))\geq f(u_{b}^{+}(t)), \end{gathered}\label{e3.2}$$ $$\begin{gathered} \text{either}\quad u(1-,t)= u_{r}^{+}(t)\\ \text{or}\quad f'(u(1-,t))\geq 0 \text{ and } f(u(1-,t))\geq f(u_{r}^{+}(t)). \end{gathered}\label{e3.3}$$ 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 $$\text{if f'(u(0+,t))>0, then v(0+,t)=v_l(t).} \label{e3.4}$$ Similarly the weak form of the boundary condition at$x=1$is $$\text{if f'(u(1-,t))<0, then v(1-,t)=v_r(t).} \label{e3.5}$$ 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 $$J(\beta)=-\int_{\{s:\beta(s)=0\}}f(u_l^+ (s))ds -\int_{\{s:\beta(s)=1\}}f(u_r^- (s))ds +\int_{\{s:0<\beta(s)<1\}}f^*(\frac{d\beta}{ds})ds. \label{e3.6}$$ 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 $$A(x,y,t) = J(\beta_0),\quad B(x,y,t)=\min_{\beta \in C^*(x,y,t)} J(\beta) \label{e3.7}$$ 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} \begin{thebibliography}{00} \bibitem{b1} C. Bardos, A. Y. Leroux and J. C. Nedelec; First order quasi-linear equations with boundary conditions, {\it Comm. Part. Diff. Eqn} {\bf 4} (1979), 1017-1034. \bibitem{j1} K. T. Joseph; A Riemann problem whose viscosity solution contain$\delta\$-measures. {\it Asym. Anal.} {\bf 7}(1993) 105-120. \bibitem{j2} K. T. Joseph and G. D. Veerappa Gowda; Explicit formula for the solution of convex conservation laws with boundary condition, {\it Duke Math. J.} {\bf 62} (1991) 401-416. \bibitem{j3} K. T. Joseph and G. D. Veerappa Gowda; Solution of convex conservation laws in a strip, {\it Proc. Indian Acad. Sci. (Math. Sci),} {\bf 102} (1992) 29-47. \bibitem{j4} K. T. Joseph; One-dimensional adhesion model for large scale structures, {\it Electronic J. Diff. 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