\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 157, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/157\hfil Explicit solutions for a system of PDEs] {Explicit solutions for a system of first-order partial differential equations} \author[K. T. Joseph\hfil EJDE-2008/157\hfilneg] {Kayyunnapara Thomas Joseph} \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 November 1, 2008. Published November 20, 2008.} \subjclass[2000]{35A20, 35L50, 35R05} \keywords{First order equations; boundary conditions; exact solutions} \begin{abstract} In this paper we construct explicit weak solutions of a system of two partial differential equations in the quarter plane $\{(x,t):x>0,t>0\}$ with initial conditions at $t=0$ and a weak form of Dirichlet boundary conditions at $x=0$. This system was first studied by LeFloch \cite{le1}, where he constructed explicit formula for the weak solution of pure initial value problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \section{Introduction} LeFloch \cite{le1} constructed an explicit formula for the solution to initial-value problem $$\begin{gathered} u_t + f(u)_x =0,\\ v_t + f'(u) v_x =0, \end{gathered} \label{e1.1}$$ with initial conditions $$\begin{pmatrix} u(x,0)\\ v(x,0) \end{pmatrix} = \begin{pmatrix} u_0(x)\\ v_0(x) \end{pmatrix}, \label{e1.2}$$ in the domain $\{(x,t) : -\infty < x < \infty, t>0 \}$, where $f(u)$ is strictly convex. The first equation is a convex conservation law and the Lax formula \cite{la1} gives the entropy weak solution $u(x,t)$ when the initial data $u(x,0)=u_0(x)$ is in the space of bounded measurable functions. The solution $u(x,t)$ remains in the space bounded functions and is locally a $BV$ function for $t>0$. Then the second equation for $v$ is a nonconservative scalar equation with bounded and $BV_{loc}$ function $f'(u)$ as coefficient and LeFloch \cite{le1} gave an explicit formula for the solution $v(x,t)$ satisfying initial data $v(x,0)=v_0(x)$, when $v_0$ is Lipschitz continuous. To justify the nonconservative product which appear in the second equation Volpert product \cite{v1} was used and the second equation was interpreted in the sense of measures. In this paper we study \eqref{e1.1} in the quarter plane $\{(x,t) : x >0, t>0 \}$, supplemented with an initial condition at $t=0$ $$\begin{pmatrix} u(x,0)\\ v(x,0) \end{pmatrix} = \begin{pmatrix} u_0(x)\\ v_0(x) \end{pmatrix} \label{e1.3}$$ and a weak form of the Dirichlet boundary condition, $$\begin{pmatrix} u(0,t)\\ v(0,t) \end{pmatrix} = \begin{pmatrix} u_b(t)\\ v_b(t) \end{pmatrix} \label{e1.4}$$ where $u_0(x)$ is bounded measurable and $v_0(x)$ are Lipschitz continuous functions of $x$ and $u_b(t)$ and $v_b(t)$ are Lipschitz continuous functions of $t$. Indeed with strong form of Dirichlet boundary conditions \eqref{e1.4}, there is neither existence nor uniqueness as the speed of propagation $\lambda =f'(u)$ depends on the unknown variable $u$ 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 (see LeFloch \cite{le2}): $$\begin{gathered} \text{either } u(0+,t)= u_{b}^{+}(t)\\ \text{or } f'(u(0+,t))\leq 0 \text{ and } f(u(0+,t))\geq f(u_{b}^{+}(t)). \end{gathered}\label{e1.5}$$ Here $u_{b}^{+}(t)= \max\{u_b(t),\lambda\}$ where $\lambda$ is the unique point where $f'(u)$ changes sign. Because of convexity of $f$, $f(\lambda)=\inf{f(u)}$. 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{e1.5} by Joseph and Gowda \cite{j3} and LeFloch \cite{le2}. 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 $f'(u(x,t))$. Now $v(0+,t) =v_b(t)$ 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 is $$\text{if f'(u(0+,t))>0, then v(0+,t)=v_b(t).} \label{e1.6}$$ The aim of this paper is to construct explicit formula for \eqref{e1.1}, with initial condition \eqref{e1.3} and boundary conditions \eqref{e1.5} and \eqref{e1.6}. We also indicate some generalizations to some other systems. The question of uniqueness is under investigation. \section{A formula for the solution} In this section, using the explicit formula derived in \cite{j1,j3} for the scalar convex conservation laws with initial condition and Bardos Leroux and Nedelec boundary condition \eqref{e1.6}, we construct a solution for the problem stated in the introduction. To be more precise, We assume $f(u)$ satisfies the following conditions $$f''(u)>0, \quad \lim_{u \to \infty}\frac{f(u)}{u} = \infty, \label{e2.1}$$ and let $f^{*}(u)$ be the convex dual of $f(u)$ namely, $f^{*}(u)= \max_{\theta}[\theta u -f(\theta)]$. 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 flux $f(u)$ and boundary data $u_b(t)$, we define the functional $J(\beta)$ on $C(x,y,t)$ $$J(\beta) = -\int_{\{s:\beta(s)=0\}}f(u_B(s)^{+})ds + \int_{\{s:\beta(s) \neq 0\}}f^{*}\big(\frac{d\beta(s)}{ds}\big)ds. \label{e2.2}$$ 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 $$A(x,y,t)= J(\beta_0) =t f*\big(\frac{x-y}{t}\big). \label{e2.3}$$ 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{e2.2} that $$J(\beta) = J(x,y,t,t_1,t_2) = -\int_{t_1}^{t_2}f(u_B(s)^{+})ds + t_1 f^{*}(\frac{y}{-t_1})+(t-t_2) f^{*}\big(\frac{x}{t-t_2}\big). \label{e2.4}$$ 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 $(x,t)$. $J(\beta) = J(x,y,t,t_1=0,t_2) = -\int_{0}^{t_2}f(u_B(s)^{+})ds + (t-t_2) f^{*}(\frac{x}{t-t_2}).$ 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{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{e2.5} is a Lipschitz continuous so that \begin{aligned} Q(x,y,t)&= \min\{J(\beta) : \beta \in C(x,y,t)\}\\ & = \min \{A(x,y,t),B(x,y,t)\}, \end{aligned} \label{e2.6} and $$U(x,t)= \min \{Q(x,y,t) + U_0(z), \,\, 0\leq y< \infty\} \label{e2.7}$$ are Lipschitz continuous functions in their variables, where $U_0(y)=\int_0^y u_0(z)dz$. Further minimum in \eqref{e2.7} is attained at some value of $y\geq 0$ which depends on $(x,t)$, we call it $y(x,t)$. If $A(x,y(x,t),t)\leq B(x,y(x,t),t)$ $$U(x,t)=tf^{*}(\frac{x-y(x,t)}{t}) + U_0(y), \label{e2.8}$$ and if $A(x,y(x,t),t)>B(x,y(x,t),t)$ $$U(x,t)=J(x,y(x,t),t,t_1(x,y(x,t),t),t_2(x,y(x,t),t)) + U_0(y). \label{e2.9}$$ Here and hence forth $y(x,t)$ is a minimizer in \eqref{e2.7} and in the case of \eqref{e2.9}, $t_2(x,t)=t_2(x,y(x,t),t)$ and $t_1(x,t)=t_1(x,y(x,t),t)$. \begin{theorem}\label{thm2.1} For every $(x,t)$ minimum in \eqref{e2.7} 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 points $(x,t)$ satisfying $U(x,t)=A(x,y(x,t),t)\leq B(x,y(x,t),t)$, define $$\begin{gathered} u(x,t)= (f^{*})'(\frac{x-y(x,t)}{t})\\ v(x,t)= v_0(y(x,t)). \end{gathered} \label{e2.10}$$ and for the points $(x,t)$ where $B(x,y(x,t),t)0$, they are nondecreasing function of $x$ and hence except for a countable number of points they are equal. Corresponding $t_2^{-}(x,t)$ and $t_2^{+}(x,t)$ have the following properties. They are nondecreasing function of $x$, for each fixed $t$ and except for a countable number of points $x$ they are equal and nondecreasing function of $t$, for each fixed $x$ and except for a countable number of points $t$ they are equal. Further if $A(x,y(x,t),t)B(x,y(x,t),t)$, for some $x=x_0$ then this continues to be so for all $x>x_0$. It was proved in \cite{j3}, that $u(x,t)=Q_1(x,y(x,t))=\partial_{x}U(x,t)$ where $Q_1(x,y,t)= \partial_x Q(x,y,t)$, is the weak solution of $$u_t+f(u)_x=0 \label{e2.12}$$ satisfying the initial condition $u(x,0)=u_0(x)$ and weak form of boundary condition \eqref{e1.5}. To show that $v$ satisfies the second equation, we follow LeFloch \cite{le1} and use the nonconservative product of Volpert \cite{v1} in sense of measures. Since $u$ is a function of bounded variation, we write $[0,\infty)\times[0,\infty) =S_c \cup S_j \cup S_n$ where $S_c$ and $S_j$ are points of approximate continuity of $u$ and points of approximate jump of $u$ and $S_n$ 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 $g(u)v_x$ is defined as a Borel measure in the following manner. Consider the averaged superposition of $g(u)$ (see Volpert \cite{v1}) $$\overline{g(u)}(x,t) = \begin{cases} g(u(x,t)),&\text{if } (x,t) \in S_c,\\ \int_0^1 g(1-\alpha)(u(x-,t)+\alpha u(x+,t))d\alpha, &\text{if }(x,t) \in S_j \end{cases} \label{e2.13}$$ and the associated measure $$[g(u)v_x](A)=\int_{A}\overline{g(u)}(x,t)v_x \label{e2.14}$$ where $A$ is a Borel measurable subset of $S_c$ and $$[g(u)v_x](\{(x,t)\})=\overline{g(u)}(x,t)(v(x+0,t)-v(x-0,t)) \label{e2.15}$$ provided $(x,t) \in S_j$. The second equation in \eqref{e1.1} is understood as $$\mu = v_t + \overline{f'(u)}(u)v_x=0 \label{e2.16}$$ in the sense of measures. Let $(x,t)\in S_c$ and $u={f^{*}}'(\frac{x-y(x,t)}{t})$, since $u$ satisfies \eqref{e2.12}, we have \begin{aligned} f''(u)\{-\frac{(x-y(x,t))}{t^2}-\frac{\partial_{t}y(x,t)}{t}+ f'(u)\frac{(1-\partial_{x}y(x,t))}{t}\}=0. \end{aligned} This can be written as $$f''(u)\{-\frac{1}{t}[(\frac{(x-y(x,t))}{t}-f'(u))]-\frac{1}{t}[\partial_{t}y(x,t)+ f'(u)\partial_{x}y(x,t)]\}=0. \label{e2.17}$$ Using $f''(u)>0$ and $f'(u)$ and $(f^{*}{'})(u)$ are inverses of each other, it follows from \eqref{e2.17} that $$\partial_{t}y(x,t) + f'(u)\partial_{x}y(x,t)=0. \label{e2.18}$$ Now $\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=(\frac{dv_0}{dx})(y(x,t) \{\partial_{t}y(x,t) + f'(u)\partial_{x}y(x,t)\}$ and from \eqref{e2.18}, we get $$\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=0. \label{e2.19}$$ Similarly if $(x,t)\in S_c$ and $u(x,t)={f^{*}}'(\frac{x}{t-t_2(x,y(x,t),t)})$, we can show that $\partial_t(t_2(x,y(x,t),t)) + f'(u(x,t))\partial_x(t_2(x,y(x,t),t) =0$ and hence $$\partial_{t}v(x,t) + f'(u)\partial_{x}v(x,t)=0, \label{e2.20}$$ So from \eqref{e2.19} and \eqref{e2.20}, for any Borel subset $A$ of $S_c$ $$\mu(A)=0. \label{e2.21}$$ Now we consider a point $(s(t),t) \in S_j$, then $\frac{ds(t)}{dt} = \frac{f(u(s(t)+,t))-f(u(s(t)-,t))}{u(s(t)+,t)-u(s(t)-,t)}$ is the speed of propagation of the discontinuity at this point. \begin{aligned} &\mu\{(s(t),t)\}\\ &=-\frac{ds(t)}{dt}(v(s(t)+,t)-v(s(t)-,t))\\ &\quad +\int_0^1 f'(u(s(t)-,t) +\alpha (u(s(t)+,t) - u(s(t)-,t))d\alpha (v(s(t)+,t)-v(s(t)-,t))\\ &=[-\frac{ds(t)}{dt} + \frac{f(u(s(t)+,t))-f(u(s(t)-,t))}{u(s(t)+,t)-u(s(t)-,t)}] (v(s(t)+,t)-v(s(t)-,t))\\ &=0. \end{aligned} \label{e2.22} Form \eqref{e2.21} and \eqref{e2.22}, \eqref{e2.16} follows. To show that the solution satisfies the initial conditions, first we observe that given $\epsilon>0$ there exists $\delta>0$ such that for all $x\geq\epsilon$, $t\leq \delta$, $u(x,t)=(f^{*})'(\frac{x-y(x,t)}{t})$ where $y(x,t)$ minimizes $\min_{y\geq 0}[U_0(y)+t f^{*}(\frac{x-y}{t})]$ see \cite{j3}. So $u$ and $v$ are given by the formula \eqref{e2.10}. Then Lax's argument \cite{la1}, gives $\lim_{t\to 0}u(x,t)=u_0(x)$ a.e. $x\geq \epsilon$. Since $\epsilon>0$ is arbitrary, $\lim_{t\to 0}u(x,t)=u_0(x),\,\,\, a.e.\,\,x.$ Since $f'$ and ${f^{*}}'$ are inverses of each other $y(x,t)-x = -t f'(u(x,t))$, then it follows that $y(x,t)\to x$ as $t\to 0$ a.e $x$. Since $v_0$ is continuous we get $\lim_{t\to 0}v(x,t)=\lim_{t\to 0}v_0(y(x,t))=v_0(x),\,\,\, a.e.\,\,x.$ Now we show the solution satisfies the boundary condition \eqref{e1.5} and \eqref{e1.6}. That the $u$ component satisfies the boundary condition \eqref{e1.5} is proved in \cite{j3}. Further if $f'(u(0+,t))>0$ then $f'(u(x,t))>0$ for $00$. So we have $\lim_{x\to 0} v(x,t)=\lim_{x\to 0}v_b(t_2(x,t))=v_b(t).$ as $v_b$ is continuous. This proves $v$ satisfies the boundary condition \eqref{e1.6}. The proof of the theorem is complete. \end{proof} \section{Extensions to some other cases} \subsection*{Generalized Lax equation} The initial value problem for the system $$\begin{gathered} u_t +(\log(a e^u +b e^{-u}))_x =0\\ v_t + \frac{ae^u-be^{-u}}{ae^u+be^{-u}}v_x=0 \end{gathered} \label{e3.1}$$ was studied and explicit solution was constructed by Joseph and Gowda \cite{j5} using a difference scheme of Lax \cite{la1}. This system of equations is of the form \eqref{e1.1},with $$f(u)=\log(ae^u + be^{-u}) \label{e3.2}$$ For the case $f(u)$ satisfying \eqref{e2.1}, $f^{*}$ is defined everywhere. The flux $f(u)$ given by \eqref{e3.2} is convex but does not satisfies \eqref{e2.1} and $f^{*}$ is not defined everywhere. Indeed $f^{*}$ is defined only on $(-1,1)$ and is given by $$f^{*}(u)=\frac{1}{2}\log\begin{pmatrix}(1+u)^{1+u} (1-u)^{1-u}\end{pmatrix} -\frac{1}{2}\log\begin{pmatrix}4 a^{1+u} b^{1-u}\end{pmatrix} \label{e3.3}$$ and its derivative is $${f^{*}}'(u)=\frac{1}{2}\log(\frac{b}{a}\frac{1+u}{1-u}). \label{e3.4}$$ Explicit formula of the theorem (2.1) can be obtained for \eqref{e3.1} on the domain $D = \{(x,t), x>0,t>0\}$ with initial condition \eqref{e1.3} and boundary conditions \eqref{e1.5} and \eqref{e1.6} with minor modifications. Here we define $C(x,y,t)$, the set of curves $\beta$ as in section 2, but with a restriction on its slope $|\frac{d\beta(s)}{ds}|<1$. Using the same notations as in theorem, and using the explicit form of ${f^{*}}'(u)$ given by \eqref{e3.4}, we have the following result. \begin{theorem}\label{thm3.1} For every $(x,t)$, $x>0$, $t>0$, let $(u,v)$ be defined as follows: When $A(x,y(x,t),t)\leq B(x,y(x,t),t)$, by $u(x,t)=\frac{1}{2}\log[\frac{b}{a}\frac{t+x-y(x,t)}{t-x+y(x,t)}],\quad v(x,t)=v_0(y(x,t));$ when $A(x,y(x,t),t)>B(x,y(x,t),t)$, by $u(x,t)=\frac{1}{2}\log[\frac{b}{a}\frac{t+x+t_2(x,t)}{t-x+t_2(x,t)}],\quad v(x,t)=v_b(t_2(x,t)).$ Then $(u,v)$ solves \eqref{e3.1}, satisfies the initial conditions \eqref{e1.3} and the boundary conditions \eqref{e1.5} and \eqref{e1.6}. \end{theorem} \subsection*{Generalized Hopf equation} Solution for the initial-value problem for the nonconservative system for $u_j, j=1,2,\dots, n$ $$(u_j)_t + (\sum_{k=1}^n c_k u_k)(u_j)_x =0, \quad j=1,2,\dots ,n \label{e3.5}$$ was constructed by Joseph \cite{j2,j4} by a vanishing viscosity method and a generalization of Hopf-Cole transformation. Here we assume that at least one $k$, $c_k\neq 0$. When $n=1,c_1=1$, \eqref{e3.5} is the inviscid Burgers equation or the Hopf equation and Hopf \cite{h1} derived a formula for the entropy weak solution for the initial value problem and boundary case was treated in \cite{j1}. In the present discussion we consider \eqref{e3.5} in $D=\{(x,t): x>0, t>0\}$ with initial condition $$u_j(x,0)=u_{0j}(x),\quad x>0,\quad j=1,2,\dots ,n \label{e3.6}$$ and boundary conditions $$u_j(0,t)=u_{bj}(t),\quad t>0\quad j=1,2,\dots ,n. \label{e3.7}$$ Here again a weak form of the boundary condition is required as characteristic speed of the system, $\sigma =\sum_{k=1}^n c_k u_k$ need not be positive at the boundary $x=0$. First we note from \eqref{e3.5} that $u_j$ satisfies $$(u_j)_t + \sigma (u_j)_x =0, \quad j=1,2,\dots ,n \label{e3.8}$$ where $\sigma$ satisfies $$\sigma_t + (\frac{\sigma^2}{2})_x =0. \label{e3.9}$$ Now \eqref{e3.9} together with \eqref{e3.8} is exactly the form of equation we have studied in section 1, with $f(u)={u^2/2}$. Let $\sigma$ is the entropy weak solution of \eqref{e3.9} with the initial condition $$\sigma(x,0)=\sigma_0(x) \label{e3.10}$$ and weak form of boundary condition $$\begin{gathered} \text{either \sigma(0+,t)= \sigma_{b}^{+}(t)}\\ \text{or \sigma(0+,t)\leq 0 and \frac{u(0+,t)}{2}\geq \frac{u_{b}^{+}(t)}{2}}, \end{gathered} \label{e3.11}$$ with $\sigma_0(x)=\sum_{k=1}^n c_k u_{0k}(x)$ and $\sigma_b(t)=\sum_{k=1}^n c_k u_{bk}(t)$ constructed in \cite{j1,j3}. The analysis of section 1 then shows that with the formulation of boundary condition $$\text{if \sigma(0+,t)>0, then u_j(0+,t)=u_{bj}(t)}. \label{e3.12}$$ for $u_j$, Theorem (1.1) applies to the present case with $f(u)=\frac{u^2}{2}$. With the same notations as Theorem 1.1, we obtain the following theorem. \begin{theorem}\label{thm3.2} For $x>0$, $t>0$, let $u_j$ be defined as follows:\\ For points $(x,t)$ where $U(x,t)=A(x,y(x,t),t)\leq B(x,y(x,t),t)$, define $u_j(x,t)= u_{0j}(y(x,t)),$ and for the points $(x,t)$ where \$B(x,y(x,t),t)