Abstract: In this note we present a periodic version of an adaptation of the Glimm scheme introduced by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973). For special classes of $2\X2$ systems of conservation laws these adaptions of the Glimm scheme give global existence of solutions of the Cauchy problem with large initial data in $L^\infty\cap BV_{loc}(\R)$, for Nishida-Bakhvalov's class, and in $L^\infty\cap BV(\R)$, in the case of DiPerna's class. 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-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 systems motivated by isentropic gas dynamics.
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