Asymptotic behavior of the solutions to a one-dimensional motion of compressible viscous fluids

summary:We study the one-dimensional motion of the viscous gas represented by the system $v_t-u_x = 0$, $ u_t+ p(v)_x = \mu(u_x/v)_x + f \left( \int_0^xv\dd x,t \right)$, with the initial and the boundary conditions $(v(x,0), u(x,0)) = (v_0(x), u_0(x))$, $u(0,t) = u(X,t) = 0$. We are concerned with the external forces, namely the function $f$, which do not become small for large time $t$. The main purpose is to show how the solution to this problem behaves around the stationary one, and the proof is based on an elementary $L^2$-energy method