\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 58, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/58\hfil Singular solutions] {Singular solutions for 2x2 systems in nonconservative form with incomplete \\ set of eigenvectors} \author[A. P. Choudhury \hfil EJDE-2013/58\hfilneg] {Anupam Pal Choudhury} % in alphabetical order \address{Anupam Pal Choudhury \newline TIFR Centre for Applicable Mathematics\\ Sharada Nagar, Chikkabommasandra, GKVK P.O.\\ Bangalore 560065, India} \email{anupam@math.tifrbng.res.in} \thanks{Submitted December 7, 2012. Published February 26, 2013.} \subjclass[2000]{35L65, 35L67} \keywords{Hyperbolic systems of conservation laws; $\delta$-shock wave type solution; \hfill\break\indent weak asymptotic method} \begin{abstract} In this article, we study the initial-value problem for two first-order systems in non-conservative form. The first system arises in elastodynamics and belongs to the class of strictly hyperbolic, genuinely nonlinear systems. The second system has repeated eigenvalues and an incomplete set of right eigenvectors. Solutions to such systems are expected to develop singular concentrations. Existence of singular solutions to both the systems have been shown using the method of weak asymptotics. The second system has been shown to develop singular concentrations even from Riemann-type initial data. The first system differing from the second in having an extra term containing a positive constant $k$, the solution constructed for the first system have been shown to converge to the solution of the second as $k$ tends to $0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The initial-value problem for the first-order quasilinear hyperbolic system $$\begin{gathered} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} -\frac{\partial \sigma}{\partial x}=0, \\ \frac{\partial \sigma}{\partial t}+u\frac{\partial \sigma}{\partial x} -k^2\frac{\partial u}{\partial x}=0 \end{gathered} \label{e1.1}$$ (in the domain $\Omega=\{(x,t):-\infty0\}$) arising in applications in elastodynamics, has been well studied (see \cite{c1,j1,j2,l1}). Here $k$ is a positive constant. It is a strictly hyperbolic system having two real distinct eigenvalues given by $$\lambda_{1}(u,\sigma)=u-k,\ \lambda_{2}(u,\sigma)=u+k$$ with the corresponding right eigenvectors $$E_{1}(u,\sigma)=\begin{pmatrix} 1 \\ k \end{pmatrix},\quad E_{2}(u,\sigma)=\begin{pmatrix} 1 \\ -k \end{pmatrix}.$$ Now letting $k \to 0$, we see that the eigenvalues $\lambda_{1}(u,\sigma)$ and $\lambda_{2}(u,\sigma)$ tend to coincide. In particular, taking $k=0$ in \eqref{e1.1} we arrive at the system $$\begin{gathered} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x} -\frac{\partial \sigma}{\partial x}=0, \\ \frac{\partial \sigma}{\partial t}+u\frac{\partial \sigma}{\partial x}=0. \end{gathered} \label{e1.2}$$ which has repeated eigenvalues $\lambda_{1}(u,\sigma)=\lambda_{2}(u,\sigma)=u$ and an incomplete set of right eigenvectors (we can take $\begin{pmatrix} 1\\ 0\end{pmatrix}$ to be a right eigenvector). In \cite{z1}, a class of 2x2 systems in conservative form having an incomplete set of eigenvectors everywhere has been considered. These systems exhibit development of singular concentrations. We expect a similar kind of development of singular concentration for the system \eqref{e1.2}. But the analysis in \cite{z1} cannot be applied directly due to the following two reasons: 1. The system \eqref{e1.2} is in nonconservative form and hence we need to give a suitable meaning to the nonconservative products which in general lead to different solutions depending upon the meaning attached (see \cite{d1},\cite{r1},\cite{v1}). 2. One of the assumptions that has been used in \cite{z1} is that the eigenvalue should have vanishing directional derivative along a right eigenvector. But the eigenvalue $u$ in this case has a nonvanishing directional derivative along any right eigenvector. Thus the assumption is not satisfied. Another reason to expect singular solutions for the system \eqref{e1.2} comes from studying the behaviour as $k\to 0$ of the shock and rarefaction curves obtained for the Riemann problem for the system \eqref{e1.1}. In \cite{j2}, the Riemann problem for the system \eqref{e1.1} has been studied using Volpert's product. Starting with Riemann type initial data $$(u(x,0),\sigma(x,0))=\begin{cases} (u_{L},\sigma_{L}),& x<0\\ (u_{R},\sigma_{R}),& x>0. \end{cases}$$ the shock curves $S_{1}(u_{L},\sigma_{L}),S_{2}(u_{L},\sigma_{L})$ and the rarefaction curves $R_{1}(u_{L},\sigma_{L}),$\\ $R_{2}(u_{L},\sigma_{L})$ can be written down in the $u-\sigma$ plane as in \cite{j2}: \begin{gathered} R_{1}(u_{L},\sigma_{L}):\sigma=\sigma_{L}+k(u-u_{L}),\; u>u_{L},\\ R_{2}(u_{L},\sigma_{L}):\sigma=\sigma_{L}-k(u-u_{L}),\; u>u_{L},\\ S_{1}(u_{L},\sigma_{L}):\sigma=\sigma_{L}+k(u-u_{L}),\; u0$, we get the shock curves \begin{gather*} S_{1}:[\sigma]=k[u],\\ S_{2}:[\sigma]=-k[u]. \end{gather*} Also in this case, we have$u\frac{\partial \sigma}{\partial x}=\lim_{\epsilon \to 0}u(x,t,\epsilon) \frac{\partial \sigma(x,t,\epsilon)}{\partial x}=-\sigma_{1}(u_{0} +\frac{u_{1}}{2})\delta$, which again is the Volpert's product (the negative sign arises because of the convention on$[\sigma]$). Thus we recover the results proved in \cite{j2} for the shock-wave case; see \eqref{e1.3}. \end{remark} \begin{remark}[Overcompressivity condition for$\delta$-shock wave solutions]\rm We recall that the overcompressivity condition (see \cite{a1,k1}) for the$\delta$-shock wave solutions for a$n\times n$system is $$\lambda_{k}(v_{R})<\dot \phi(t)<\lambda_{k}(v_{L}),\quad k=1,\dots,n.$$ Therefore, for the system \eqref{e1.1}, it takes the form \begin{gather*} u_{0}-k<\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}2k>0, \\ -(\frac{u_{1}}{2}-k)<\frac{\sigma_{1}}{u_{1}}<\frac{u_{1}}{2}-k. \end{gather*} \end{remark} Next we prove the existence of a weak asymptotic solution for system \eqref{e1.2}. \begin{theorem} For$t\in [0,\infty)$, the Cauchy problem \eqref{e1.2}, \eqref{e3.2} has a weak asymptotic solution \eqref{e3.3} with$\phi(t),e(t)\ and\ p(t)$given by the relations $$\begin{gathered} \dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]},\ \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}},\\ \frac{1}{2}p^2(t)\omega_{0}-e(t)=0, \end{gathered} \label{e4.7}$$ where$\omega_{0}$is a positive constant$($defined in Section$3)$. \end{theorem} \begin{proof} Proceeding as in the proof of the Theorem 4.1, we find that the smooth ansatz \eqref{e3.3} is a weak asymptotic solution provided$p(t),\phi(t),e(t)$can be solved from the equations $$\begin{gathered} \dot{\phi}(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}, \\ \dot{e}(t)=\frac{\sigma_{1}^2}{u_{1}}, \\ \frac{1}{2}p^2(t)\omega_{0}-e(t)=0. \end{gathered} \label{e4.8}$$ The ordinary differential equations for$\phi(t)$and$e(t)$can be solved with the initial conditions$\phi(0)=0$and$e(0)=e^{0}$and we have $\phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad e(t)=\frac{\sigma_{1}^2}{u_{1}}t+e^{0}.$ Next substituting$e(t)$in the last equation of \eqref{e4.8}, we can solve for$p(t)$taken in the form$p(t)=p_{1}(t)+ip_{2}(t)$and hence we have a weak asymptotic solution of system \eqref{e1.2}. \end{proof} Therefore, from the previous theorem, we have the following result. \begin{theorem} For$t\in [0,\infty)$, the Cauchy problem \eqref{e1.2}, \eqref{e3.2} has a generalised$\delta$-shock wave type solution \eqref{e3.1} with$\phi(t)$and$e(t)$given by the relations $$\phi(t)=\frac{[\frac{u^2}{2}]-[\sigma]}{[u]}t,\quad e(t)=\frac{\sigma_{1}^2}{u_{1}}t+e^{0}. \label{e4.9}$$ \end{theorem} \begin{remark} \rm The overcompressivity assumption for the system \eqref{e1.2} yields$u_{1}>0$and$-\frac{u_{1}}{2}<\frac{\sigma_{1}}{u_{1}}<\frac{u_{1}}{2}$. \end{remark} \begin{remark} \rm If$u_{1}>0$(follows from the overcompressivity condition above), then from equation \eqref{e4.7} it follows that$\dot{e}(t)>0$. If in addition we have$e^{0}=0$, then$e(t)=\frac{\sigma_{1}^2}{u_{1}}t$which is greater than zero for all t. In this case, it is sufficient to consider$p(t)$as a real-valued function only. \end{remark} \begin{remark} \rm From Remark 4.8 it also follows that we might have a singular concentration developing in the solution of the system \eqref{e1.2} even if we start with Riemann type initial data. \end{remark} \begin{remark} \rm If we take$\sigma_{0}=\sigma_{1}=0$in \eqref{e3.2}, then proceeding as in the proof of the Theorem 4.5 we obtain a generalised$\delta$-shock wave type solution for the system \eqref{e1.2} of the form: \begin{gather*} u(x,t)=u_{0}+u_{1}H(-x+\phi(t)),\\ \sigma(x,t)=e^{0}\delta(x-\phi(t)) \end{gather*} where$\dot{\phi}(t)=\frac{[\frac{u^2}{2}]}{[u]}$. \end{remark} \begin{remark}[Dependence of the solutions on$k$] \rm From the structure of the generalised solutions for systems the \eqref{e1.1}, \eqref{e1.2} obtained from Theorem 4.2 and Theorem 4.6, it is quite evident that as$k$tends to$0$, the generalised solution obtained for the system \eqref{e1.1} actually converges (in distributional limit) to that obtained for the system \eqref{e1.2}. This observation therefore justifies our motivation to study the system \eqref{e1.2} based upon the solutions of the system \eqref{e1.1} (letting$k\to 0$). \end{remark} \subsection*{Acknowledgements} The author is grateful to the anonymous referee whose constructive criticism helped to improve the contents and presentation of this article. \begin{thebibliography}{00} \bibitem{a1} Albeverio, S.; Shelkovich, V. M.; On the delta-shock front problem. \textit{Analytical approaches to multidimensional balance laws, Nova Sci. Publ., New York} (2006), 45-87 \bibitem{c1} Cauret, J. J.; Colombeau, J. F.; LeRoux, A. Y.; Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equation, \textit{J. Math. Anal. Appl.} 139 (1989), 552-573. \bibitem{d1} Dal Maso, G.; Lefloch, P. G.; Murat, F.; Definition and weak stability of non-conservative products, \textit{J. Math. Pures Appl.} 74 (1995), 483-548. \bibitem{d2} Danilov, V.G.; Mitrovic, D.; Weak asymptotic of shock wave formation process, \textit{Nonlinear Anal.} 61 (2005), 613-635. \bibitem{j1} Joseph, K. T.; Generalized solutions to a Cauchy problem for a nonconservative hyperbolic system, \textit{J. Math. Anal. Appl.} 207 (1997), 361-389. \bibitem{j2} Joseph, K. T.; Sachdev, P. L.; Exact solutions for some non-conservative hyperbolic systems, \textit{Internat. J. Non-Linear Mech.} 38 (2003), 1377-1386. \bibitem{k1} Kalisch, H.; Mitrovic, D.; Singular solutions for the shallow-water equations, \textit{IMA J. Appl. Math.} 77 (2012), 340-350. \bibitem{l1} Lefloch, P. G.; Liu, T.-P.; Existence theory for nonlinear hyperbolic systems in nonconservative form, \textit{Forum Math.} 3 (1993), 261-280. \bibitem{r1} Raymond, J. P.; A new definition of nonconservative products and weak stability results, \textit{Boll. Un. Mat. Ital. B (7)} 10 (1996), 681-699. \bibitem{v1} Volpert, A. I.; The space BV and quasilinear equations, \textit{Math. USSR Sb. 2} (1967), 225-267. \bibitem{z1} Zheng, Y.; Systems of conservation laws with incomplete sets of eigenvectors everywhere. \textit{Advances in nonlinear partial differential equations and related areas (Beijing, 1997), World Sci. Publ., River Edge, NJ.} (1998), 399-426. \end{thebibliography} \section*{Addendum posted on July 26, 2013} The author would like to make a few minor corrections, and to mention that no changes in the main results take place. Since all the corrections are on page 6, we rewrite this page from the top to the beginning of Section 4. \medskip Let$c=(\frac{1}{2}-\frac{\sigma_{1}}{u_{1}^2})$. Let us define the smooth function$H(x,\epsilon)$as follows: $$H(x,\epsilon)=\begin{cases} 0,& x\leq -4\epsilon\\ c,& -3\epsilon\leq x \leq 3\epsilon\\ 1,& x\geq 4\epsilon \end{cases} \tag{3.5}$$ and is continued smoothly in the regions$(-4\epsilon,-3\epsilon)$and$(3\epsilon,4\epsilon)$. We take$H_{u}(x,\epsilon)=H_{\sigma}(x,\epsilon)=H(x,\epsilon)$. Again a little bit of calculation shows that \begin{gather*} H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(\epsilon),\quad \frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(\epsilon),\\ H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x} =\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(\epsilon) \end{gather*} Since the supports of$R(x,\epsilon)$and$\delta(x,\epsilon)$are contained in$(-3\epsilon,3\epsilon)$, it again follows that \begin{gather*} H(x,\epsilon)\frac{\partial R(-x,\epsilon)}{\partial x}=c\frac{\partial R(-x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(\epsilon),\\ R(-x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=0.R(-x,\epsilon)=0,\\ H(x,\epsilon)\frac{\partial \delta(-x,\epsilon)}{\partial x}=c\delta'(-x)+o_{\mathcal{D}'}(\epsilon). \end{gather*} From the above discussions we then have the following lemma. \medskip \textbf{Lemma 3.1.} {\it Choosing the regularizations and corrections as in$(3.4)$and$(3.5)\$ we have the following weak asymptotic expansions: \begin{gather*} R(x,\epsilon)=o_{\mathcal{D}'}(1),\quad \frac{\partial R(x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1), \\ R^2(x,\epsilon)=\omega_{0}\delta(x)+o_{\mathcal{D}'}(1), \\ R(x,\epsilon)\frac{\partial R(x,\epsilon)}{\partial x} =\frac{1}{2}\omega_{0}\delta'(x)+o_{\mathcal{D}'}(1), \\ \delta(x,\epsilon)=\delta(x)+o_{\mathcal{D}'}(1),\quad \frac{\partial \delta(x,\epsilon)}{\partial x}=\delta'(x)+o_{\mathcal{D}'}(1), \\ R(x,\epsilon)\delta(x,\epsilon)=0,\quad R(x,\epsilon)\frac{\partial \delta(x,\epsilon)}{\partial x}=0, \\ H(x,\epsilon)=H(x)+o_{\mathcal{D}'}(1),\quad \frac{\partial H(x,\epsilon)}{\partial x}=\delta(x)+o_{\mathcal{D}'}(1), \\ H(x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x} =\frac{1}{2}\delta(x)+o_{\mathcal{D}'}(1), \\ H(x,\epsilon)\frac{\partial R(-x,\epsilon)}{\partial x}=o_{\mathcal{D}'}(1),\quad R(-x,\epsilon)\frac{\partial H(x,\epsilon)}{\partial x}=0, \\ H(x,\epsilon)\frac{\partial \delta(-x,\epsilon)}{\partial x} =c\delta'(-x)+o_{\mathcal{D}'}(1),\quad \epsilon \to 0. \end{gather*} } End of addendum. \textbf{Acknowledgement.} The author would like to thank Prof. Evgeniy Panov for bringing the corrections to the author's attention. \end{document}