\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 116, 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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/116\hfil A Riemann problem with viscosity and dispersion] {A Riemann problem with small viscosity and dispersion} \author[K. T. Joseph\hfil EJDE-2006/116\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} \date{} \thanks{Submitted July 11, 2006. Published September 26, 2006.} \subjclass[2000]{35B40, 35L65} \keywords{Elastodynamics equation; viscosity; dispersion; Riemann problem} \begin{abstract} In this paper we prove existence of global solutions to a hyperbolic system in elastodynamics, with small viscosity and dispersion terms and derive estimates uniform in the viscosity-dispersion parameters. By passing to the limit, we prove the existence of solution the Riemann problem for the hyperbolic system with arbitrary Riemann data. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \section{Introduction} In this paper first we consider the boundary-value problem, for a system of nonlinear ordinary differential equations, $$\begin{gathered} -\xi \frac {du}{d\xi} + u \frac {du}{d\xi} -\frac {d\sigma}{d\xi} = \epsilon \frac {d^2u}{d\xi^2} + \gamma \epsilon^2 \frac{d^3u}{d\xi^3}, \\ -\xi \frac {d\sigma}{d \xi} +u \frac {d\sigma}{d \xi} - k^2 \frac {du}{d \xi} = \epsilon \frac {d^2\sigma}{d \xi^2} + \gamma \epsilon^2 \frac{d^3\sigma}{d\xi^3} \end{gathered}\label{e1.1}$$ for $\xi \in [a,b]$ with boundary conditions $$\begin{gathered} u(a) = u_L, u(b)=u_R, \\ \sigma(a) = \sigma_L, \sigma(b)=\sigma_R. \end{gathered} \label{e1.2}$$ This system is the self-similar vanishing diffusion-dispersion approximation of initial value problem for the system of equations which comes in elastodynamics: $$\begin{gathered} u_t + u u_x - \sigma_x = 0,\\ \sigma_t + u \sigma_x - k^2 u_x = 0, \end{gathered} \label{e1.3}$$ with Riemann initial data $$(u(x,0),\sigma(x,0))=\begin{cases} (u_L,\sigma_L), & x <0 \\ (u_R,\sigma_R) & x >0. \end{cases} \label{e1.4}$$ Here $u$ is the velocity, $\sigma$ is the stress and $k>0$ is the speed of propagation of the elastic waves. The system \eqref{e1.3} is nonconservative, strictly hyperbolic system with characteristic speeds $$\lambda_1(u,\sigma) = u - k, \lambda_2(u,\sigma) = u + k \label{e1.5}$$ and Riemann invariants $$r_1(u,\sigma)=\sigma + k u, r_2(u,\sigma)= \sigma - k u \label{e1.6}$$ and was studied by many authors \cite{c1,j1,j2,j3,j5}, with initial datas under various conditions using differece schemes or diffusion approximations. In this paper we analyse diffusion-dispersion approximations for the Riemann problem \eqref{e1.3} and \eqref{e1.4}. First we show the existence of smooth solutions $(u^\epsilon,\sigma^\epsilon)$of the problem \eqref{e1.1}-\eqref{e1.2} and derive estimates in the space of bounded variation, uniformly in $\epsilon>0$. We do not give any restrictions on the size of the initial data. Next we study $(u^\epsilon,\sigma^\epsilon)$ as $\epsilon$ tends to $0$ and show the limit function is a weak solution to \eqref{e1.3} with the Riemann initial data \eqref{e1.4} The nonconservative product which appear in the equation \eqref{e1.3} is justified by the work of LeFloch and Tzavaras \cite{l2} on nonconservative products. \section{ Self-similar vanishing diffusion-dispersion approximation} In this section, we consider the system \eqref{e1.1} and \eqref{e1.2} and prove the existence of smooth solutions. It is more convenient to work with Riemann invariants \eqref{e1.5}. Given the data $(u_L,\sigma_L), (u_R,\sigma_R)$, we define $$\begin{gathered} r_{1L} = \sigma_L + k u_L, r_{1R} = \sigma_R + k u_R,\\ r_{2L} = \sigma_L -k u_L, r_{2R} = \sigma_R - k u_R. \end{gathered} \label{e2.1}$$ The characteristic speeds \eqref{e1.5} in terms of the Riemann invariants take the form $\lambda_1(r_1,r_2) = \frac{r_1-r_2}{2k}-k,\quad \lambda_2(r_1,r_2) = \frac{r_1-r_2}{2k}+k.$ Consider the rectangle $D = [\min(r_{1L},r_{1R}),\max(r_{1L},r_{1R})]\times [\min(r_{2L},r_{2R}),\max(r_{2L},r_{2R})],$ and consider the minimum and maximum of the eigenvalues on this square $$\lambda_j^m = \min_{D}\lambda_j(r_1,r_2), \quad \lambda_j^M = \max_{D}\lambda_j(r_1,r_2),\quad j=1,2. \label{e2.2}$$ We choose $\gamma>0$, small and the boundary points $a,b$ such that 0<\gamma<\frac{1}{4(\lambda_2^M - \lambda_1^m)}, \quad \lambda_1^m -\frac{1}{\gamma}\lambda_2^M$the limiting values of$(u,\sigma)$are$(u_L,\sigma_L)(u_R,\sigma_R)$respectively. \begin{theorem} \label{thm2.1} Under the assumptions \eqref{e2.3}, for each fixed$\epsilon>0$there exits a smooth solution$(u^\epsilon(\xi),\sigma^\epsilon(\xi))$for \eqref{e1.1} and \eqref{e1.2} satisfying the estimates \begin{gather} |u^\epsilon(\xi)| + |\sigma^\epsilon(\xi)| \leq C, \quad \int_{a}^{b}| \frac {du^\epsilon}{d \xi}|d \xi + \int_{a}^{b}| \frac {d\sigma^\epsilon}{d \xi}(\xi)|d \xi \leq C, \label{e2.4} \\ |u^\epsilon(\xi) - u_L| + |\sigma^\epsilon(\xi) -\sigma_L| \leq \frac{C}{\delta} e^\frac{-(\xi -\lambda_1^m)^2}{2\epsilon} , \quad a \leq \xi \leq\lambda_1^m-\delta \label{e2.5} \\ |u^\epsilon(\xi) - u_R| + |\sigma^\epsilon(\xi) -\sigma_R| \leq \frac{C}{\delta} e^\frac{-(\xi -\lambda_2^M)^2}{2\epsilon} , \quad \lambda_2^M+\delta \leq \xi \leq b, \label{e2.6} \end{gather} for some constant$C>0$independent of$\epsilon>0$and for$\delta>0$, small. \end{theorem} \begin{proof} For convenience of notation, in the rest of this section we drop the dependence of$\epsilon$and write$u,\sigma,r_1,r_2$ect. In terms of the Riemann invariants \eqref{e1.5}, the problem \eqref{e1.1} and \eqref{e1.2} takes the form $$\begin{gathered} -\xi \frac {dr_1}{d \xi} + \lambda_1(r_1,r_2) \frac {dr}{d \xi} = \epsilon^2 \frac {d^2 r_1}{d \xi^2} + \gamma \epsilon \frac {d^3 r_1}{d\xi^3},\\ -\xi \frac {dr_2}{d \xi} + \lambda_2(r_1,r_2) \frac {dr_2}{d \xi} = \epsilon \frac {d^2 r_2}{d \xi^2}+ \gamma \epsilon^2 \frac {d^3 r_2}{d \xi^3} \end{gathered} \label{e2.7}$$ on$[a,b]$with boundary conditions $$r_1(a) = r_{1L} ,\quad r_1(b) = r_{1R} ,\quad r_2(a) = r_{2L},\quad r_2(b) = r_{2R} \label{e2.8}$$ where$r_{1L}$,$r_{1R}$,$r_{2L}$and$r_{2R}$are given by \eqref{e2.1}. From \eqref{e1.6}, the definition of$r_1(u,\sigma), r_2(u,\sigma)$, we have $$u=\frac{r_1(u,\sigma) - r_2(u,\sigma)}{2k},\quad \sigma = \frac{r_1(u,\sigma) + r_2(u,\sigma)}{2}.$$ Then to prove the theorem, it is sufficient to prove the the existence of$r_1,r_2$solution of \eqref{e2.7} and \eqref{e2.8}, with following estimates $$\begin{gathered} r_1(\xi) \in [\min(r_{1L}, r_{1R}),\; \max(r_{1L},r_{1R})], \quad \xi \in [a,b],\\ r_2(\xi) \in [\min(r_{2L}, r_{2R}), \;\max(r_{2L},r_{2R})], \quad \xi \in [a,b]; \end{gathered} \label{e2.9}$$ $$\begin{gathered} |r_1(\xi) - r_{1L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_1^m)^2}{2\epsilon}\big) ,\quad a\leq \xi \leq \lambda_1^m - \delta,\\ |r_2(\xi) - r_{2L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_2^m)^2}{2\epsilon}\big) ,\quad a \leq \xi \leq \lambda_2^m - \delta; \end{gathered} \label{e2.10}$$ $$\begin{gathered} |r_1(\xi) - r_{1R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_1^M)^2}{2\epsilon}\big) , \quad \lambda_1^M + \delta \leq \xi \leq b,\\ |r_2(\xi) - r_{2R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_2^M)^2}{2\epsilon} \big), \quad \lambda_2^M + \delta \leq \xi \leq b; \end{gathered} \label{e2.11}$$ $$\int_{a}^{b}| \frac {dr_1}{d \xi}|d \xi \leq |r_{1R}-r_{1L}|,\quad \int_{a}^{b}| \frac {dr_2}{d \xi}|d \xi \leq |r_{2R}-r_{2L}|. \label{e2.12}$$ To prove this we reduce \eqref{e2.7} and \eqref{e2.8} to an integral equation using some ideas of LeFloch and Rohde \cite{l1} and Joseph and LeFloch \cite{j4} and use a fixed point argument. First note that \eqref{e2.7} can be written in the form $$\begin{gathered} \gamma \epsilon^2 \frac{d^3r_1}{d\xi^3} + \epsilon\frac{d^2r_1}{dx^2} -(\lambda_1(r_1,r_2) - \xi) \frac{dr_1}{d\xi} =0,\\ \gamma \epsilon^2 \frac{d^2r_2}{d\xi^2} + \epsilon\frac{d^2r_2}{dx^2} -(\lambda_2(r_1,r_2) - \xi)\frac{dr_2}{d\xi} = 0. \end{gathered} \label{e2.13}$$ For$j=1,2$, let $$\varphi_1(\xi) = \frac{dr_1}{d\xi},\quad \varphi_2(x)=\frac{dr_2}{d\xi} \label{e2.14}$$ Then from \eqref{e2.13} we get $$\gamma \epsilon^2 \varphi_i''+ \epsilon \varphi_i' - (\lambda_i(r_1(\xi),r_2(\xi))-\xi) \, \varphi_i = 0. \label{e2.15}$$ Suppose we are given$r_1,r_2$smooth functions, taking values in the rectangle$D$and of finite total variation independent of$\epsilon$, \eqref{e2.15} is a second order linear ordinary differential equation for$\varphi_i$. Under the transformation, $H_i=e^{\frac{-1}{2\epsilon\gamma}\xi} \varphi_i$ \eqref{e2.15} reduces to $$H_i''= \frac{\mu_i(y)}{\gamma\epsilon^2}H \label{e2.16}$$ where $$\mu_i(\xi)=\frac{1}{4\gamma} +(\lambda_i(r_1(\xi),r_2(\xi))-\xi). \label{e2.17}$$ By taking$\gamma>0$small, we have$\mu_i(y)>0$and we can use the theorem of Olver \cite{o1} to construct solutions to \eqref{e2.15}. Indeed LeFloch and Rohde \cite{l1} showed the existence of$\varphi_i(\xi), i=1,2$satisfying the following properties: $$0<\varphi_i(\xi)\leq C/\epsilon, \int_a^b \varphi_i(\xi) d\xi =1 \label{e2.18}$$ and $$\varphi_i(\xi)\leq \begin{cases} \frac{C}{\epsilon} \exp\big(\frac{-c(x-\lambda_1^m)^2}{\epsilon}\big), & a \leq \xi \leq \lambda_1^m, \\ \frac{C}{\epsilon} \exp\big(\frac{-c(x-\lambda_1^m)^2}{\epsilon}\big), & \lambda_2^M \leq \xi \leq b, \end{cases} \label{e2.19}$$ Integrating once \eqref{e2.14} and using \eqref{e2.18} and the boundary conditions, we get, $$\begin{gathered} r_1^\epsilon(\xi) = r_{1L} + (r_{1R} - r_{1L})\int_{a}^{\xi} \varphi_1(y) dy,\\ r_2^\epsilon(\xi) = r_{2L} + (r_{2R} - r_{2L})\int_{a}^{\xi} \varphi_2(y) dy. \end{gathered} \label{e2.20}$$ It follows that to solve \eqref{e2.7} and \eqref{e2.8} with estimates \eqref{e2.4}--\eqref{e2.6}, it is enough to solve \eqref{e2.20}. To solve \eqref{e2.20}, we use the Schauder fixed point theorem applied to the function $F(r_1,r_2)(\xi) = (F_1(r_1,r_2)(\xi),F_2(r_1,r_2)(\xi))$ where $$\begin{gathered} F_1(r_1,r_2)(\xi) = r_{1L} + (r_{1R} - r_{1L})\int_{a}^{\xi} \varphi_1(y) dy,\\ F_2(r_1,r_2)(\xi) = r_{2L} + (r_{2R} - r_{2L})\int_{a}^{\xi} \varphi_2(y) dy. \end{gathered} \label{e2.21}$$ From \eqref{e2.18} and \eqref{e2.21}, it is clear that$F_1(r,s)$is a convex combination of$r_{1L}$and$r_{1R}$and$F_2(r,s)$is a convex combination of$r_{2L}$and$r_{2R}$. So the estimate $$\begin{gathered} F_1(r_1,r_2)(\xi) \in [\min(r_{1L},r_{1R}),\max(r_{1L},r_{1R})],\\ F_2(r_1,r_2)(\xi) \in [\min(r_{2L},r_{2R}),\max(r_{2L},r_{2R})] \end{gathered} \label{e2.22}$$ easily follows. Also from \eqref{e2.18} and \eqref{e2.21}, we get for$j=1,2$$$|\frac {d F_j(r,s)}{d \xi}(\xi)| \leq \frac{C}{\epsilon}. \label{e2.23}$$ Further, from \eqref{e2.19}, we get: $|F_1(r,s)(\xi) - r_{1L}| \leq \frac{C}{\epsilon}\int_{a}^\xi \exp\big(\frac{-(s -\lambda_1^m)^2}{2\epsilon}\big) ds = \frac{C \sqrt{2 \epsilon}}{\epsilon} \int_\frac{a-\lambda_1^m}{\sqrt{2\epsilon}}^\frac{(\xi-\lambda_1^m)} {\sqrt{2 \epsilon}}e^{-s^2 } ds,$ for$\quad a \leq \xi \leq \lambda_1^m$; $|F_2(r,s)(\xi) - r_{2L}| \leq \frac{C}{\epsilon}\int_{0}^\xi \exp\big(\frac{-(s -\lambda_2^m)^2}{2\epsilon}\big) ds = \frac{C \sqrt{2 \epsilon}} {\epsilon}\int_\frac{a -\lambda_2^m}{\sqrt{2 \epsilon}}^{ \frac{(\xi-\lambda_2^m)}{\sqrt{2 \epsilon}}} e^{-s^2} ds,$ for$a \leq \xi \leq \lambda_2^m$; $|F_1(r,s)(\xi) - r_{1R}| \leq \frac{C}{\epsilon}\int_\xi^{b} \exp\big(\frac{-(s -\lambda_k^M)^2}{2\epsilon}\big) ds = \frac{C \sqrt{2 \epsilon}} {\epsilon}\int_{\frac{(\xi-\lambda_1^M)}{\sqrt{2 \epsilon}}}^{\frac {b-\lambda_1^M}{\sqrt{2 \epsilon}}} e^{-s^2} ds,$ for$\lambda_1^M \leq \xi \leq b$; $|F_2(r_1,r_2)(\xi) - r_{2R}| \leq \frac{C}{\epsilon}\int_\xi^{b} \exp\big(\frac{-(s -\lambda_k^M)^2}{2\epsilon}\big) ds = \frac{C \sqrt{2 \epsilon}} {\epsilon}\int_{\frac{(\xi-\lambda_2^M)}{\sqrt{2 \epsilon}}}^{\frac {b-\lambda_2^M}{\sqrt{2 \epsilon}}} e^{-s^2} ds,$ for$\lambda_2^M \leq \xi \leq b$. Now using the asymptotic expansion $\int_y^\infty e^{-y^2} dy = (\frac{1}{2y} -O(\frac{1}{y^2}))e^{-y^2}, \quad y \to \infty$ in the above inequalities, we get $$\begin{gathered} |F_1(r_1,r_2)(\xi) - r_{1L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_1^m)^2}{2\epsilon}\big) ,\quad a\leq \xi \leq \lambda_1^m - \delta,\\ |F_2(r_1,r_2)(\xi) - r_{2L}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_2^m)^2}{2\epsilon}\big) , \quad a\leq \xi \leq \lambda_2^m - \delta; \end{gathered} \label{e2.24}$$ $$\begin{gathered} |F_1(r_1,r_2)(\xi) - r_{1R}| \leq \frac{C}{\delta}\exp\big(\frac{-(\xi -\lambda_1^M)^2}{2\epsilon}\big) , \quad \lambda_1^M + \delta \leq \xi \leq b,\\ |F_2(r_1,r_2)(\xi) - r_{2R}| \leq \frac{C}{\delta} \exp\big(\frac{-(\xi -\lambda_2^M)^2}{2\epsilon}\big) ,\quad \lambda_2^M + \delta \leq \xi \leq b. \end{gathered} \label{e2.25}$$ The estimates \eqref{e2.22} and \eqref{e2.23} show that$F=(F_1,F_2)$is compact and maps the convex set$\{(r_1,r_2)\in C[a,b]\times C[a,b] : (r_1(\xi),r_2(\xi))\in D\}$into itself, where D is the rectangle$D=[\min(r_{B},r_{R}),\max(r_{B},r_{R})]\times [\min(s_{B},s_{R}), \max(s_{B},s_{R})]$, and$C[a,b]$is the space of continuous functions with uniform norm. So by Schauder fixed point theorem there exists$(r_1,r_2)$such that$F(r_1,r_2)=(r_1,r_2)$, satisfies \eqref{e2.20} and hence is a smooth solution to \eqref{e2.7} with boundary conditions \eqref{e2.8}. Further the estimates \eqref{e2.9}-\eqref{e2.12} follows from \eqref{e2.20}, \eqref{e2.22}, \eqref{e2.24}, \eqref{e2.25} and the fact that$F(r_1,r_2)=(r_1,r_2)$. The proof of the theorem is complete. \end{proof} \section {Passage to the limit as$\epsilon \to 0$; the Riemann problem.} Here we construct solution of the Riemann problem $$\begin{gathered} u_t + u u_x - \sigma_x = 0,\\ \sigma_t + u \sigma_x - k^2 u_x = 0, \end{gathered} \label{e3.1}$$ with Riemann type initial data $$(u(x,0),\sigma(x,0))=\begin{cases} (u_L,\sigma_L) & x <0 \\ (u_R,\sigma_R) & x >0. \end{cases} \label{e3.2}$$ Since the Riemann problem is invariant under scaling, solution is sought in the form$(u(\xi,\sigma(\xi))$with$\xi=x/t$. Then \eqref{e3.1} and \eqref{e3.2} takes the form $$\begin{gathered} -\xi \frac {du}{d\xi} + u \frac {du}{d\xi} -\frac {d\sigma}{d\xi} = 0 \\ -\xi \frac {d\sigma}{d \xi} +u \frac {d\sigma}{d \xi} - k^2 \frac {du}{d \xi} = 0 \end{gathered} \label{e3.3}$$ for$\xi \in (-\infty,\infty)$with boundary conditions $$\begin{gathered} u(-\infty) = u_L, \quad u(\infty)=u_R, \\ \sigma(-\infty) = \sigma_L, \quad \sigma(\infty)=\sigma_R. \end{gathered} \label{e3.4}$$ The smooth solution$(u^\epsilon,\sigma^\epsilon)$of \eqref{e1.1} and \eqref{e1.2} constructed in the previous section is an approximation to the problem \eqref{e3.3} and \eqref{e3.4}. Because of the estimates \eqref{e2.4}, by compactnes, there exists a subsequence$(u^{\epsilon_n},\sigma^{\epsilon_n})$converges to a BV function$(u,\sigma)$as$\epsilon_n \to 0$. This limit function is not in general continuous and hence the nonconservative product which appear in the equation \eqref{e3.3} does not make sense in the theory of distribution. So we use the theory developed by LeFloch and Tzavaras \cite{l2} for nonconservative products. For completeness we briefly describe in short their results that we use. Let$u_n : [a,b] \to R^n$be a sequence of continuous functions uniformly bounded total variation: $$\sup|u_n| +TV(u_n) \leq C. \label{e3.5}$$ where$TV(u)$denotes the total variation of$u$on$[a,b]$. Define the Radon measure $\langle \mu_n,g \rangle = \int_{[a,b]} g(u_n) du_n , g \in C(R^n)$ We have $|\langle \mu_n,g \rangle | \leq TV(u_n).\sup_{|\lambda|\leq C}|g(\lambda)|.$ So by weak* compactness of$\mu_n$, there exists a subsequence$n_k$and a measure$\mu \in M(R^n)$such that $\mu_{nk} \to \mu$ in weak*$M(R^n)$. To characterize$\mu$, we need the notion of generalized graph of$u$. \begin{definition} \label{def3.1} \rm Generalized graph of$u$is defined as a Lipschitz continuous map $({X}, {U}):[0,1]\to [a,b]\times R^n$ such that \begin{itemize} \item[(a)]$({X}(0),{U}(0))=(a,u(a)),({X}(1),{U}(1))=(b,u(b))$\item[(b)]${X}$is increasing :$s_1