Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas

Abstract

This paper studies an initial boundary value problem for a one-dimensional isentropic model system of compressible viscous gas with large external forces, represented by v t −u x =0,u t +(av −γ) x =μ(u x /v) x +f(∫ 0 vdx,t), with (v(x, 0),u(x, 0))= (v 0(x),u 0(x)),u(0,t)=u(1,t)=0. Especially, the uniform boundedness of the solution in time is investigated. It is proved that for arbitrary large initial data and external forces, the problem uniquely has an uniformly bounded, global-in-time solution with also uniformly positive mass density, provided the adiabatic constant γ(>1) is suitably close to 1. The proof is based on L 2-energy estimates and a technique used in [9].