Periodic solutions of conservation laws constructed through Glimm scheme

Abstract:

We present a periodic version of the Glimm scheme applicable to special classes of $2\times 2$ systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of $2\times 2$ systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in $L^\infty \cap BV_{loc}(\mathbb {R})$, for Bakhvalov’s class, and in $L^\infty \cap BV(\mathbb {R})$, in the case of DiPerna’s class. It may also happen that the system is in Bakhvalov’s class only at a neighboorhood $\mathcal {V}$ 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 $L^\infty \cap BV(\mathbb {R})$ with $\text {TV} (U_0)<\text {const.}$, 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 (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida’s solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the $p$-systems with $p(v)=\gamma v^{-\gamma }$, $-1<\gamma <+\infty$, $\gamma \ne 0$, which, for $\gamma \ge 1$, model isentropic gas dynamics in Lagrangian coordinates.