SPATIALLY PERIODIC SOLUTIONS FOR GAS FLOWS WITH PRESSURE DEPENDING ON A VARIABLE COEFFICIENT

Abstract

We study the global existence of spatially periodic solutions for certain models of gas flow in Lagrangian coordinates for which the pressure has the form [Formula: see text], where v, as usual, is the specific volume, and [Formula: see text], [Formula: see text] are smooth functions of the variable coefficient [Formula: see text], which is assumed to satisfy suitable smoothness and decay properties, in particular, [Formula: see text], uniformly, as t →0. One important feature of our analysis is that the initial total variation over one period may be taken as large as we wish as long as [Formula: see text] is sufficiently close to 1. We also prove a non-homogeneous entropy inequality which implies the decay of the solution to the mean value as t → ∞.