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

Abstract: 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.

Keywords: compressible viscous gas, asymptotic behaviour of the solutions

Classification (MSC91): 35Q30, 76N10, 76N15

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