Preprint A50/2001
Periodic Solutions of Conservation Laws Constructed Through Glimm Scheme
Hermano Frid
Keywords: Glimm's scheme | periodic solutions | decay of periodic solutions | conservation laws
We present a periodic version of the Glimm scheme applicable to special classes of $2\X2$ systems for which it is available a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973). For these special classes of $2\X2$ systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in $L^\infty\cap BV_{loc}(\R)$, for Bakhvalov's class, and in $L^\infty\cap BV(\R)$, in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood $\Nu$ of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in $L^\infty\cap BV(\R)$ with $\TV(U_0)<\text{const.}$, for some constant which is $O((\gamma-1)^{-1})$ for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, $u(\cdot ,t)$, are uniformly bounded in $L^\infty\cap BV([0,\ell])$, for all $t>0$, where $\ell$ is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in \cite{CF} combined with a compactness theorem of DiPerna in \cite{Di83}. The question about the decay of Nishida's solution was proposed by Glimm-Lax \cite{GL} and remained open since then. The classes considered include the $p$-systems with $p(v)=\gamma v^{-\gamma}$, $-1<\gamma<+\infty$, $\gamma\ne0$, which, for $\gamma\ge 1$, model isentropic gas dynamics in Lagrangian coordinates.