Asymptotic behavior toward viscous shock for the outflow problem of barotropic Navier-Stokes equations

Abstract

We study the time-asymptotic stability of viscous shock profile for the outflow problem of the barotropic Navier-Stokes equations on the half-line. We consider the case where the far-field state, as the right end state of the 2-Hugoniot curve, belongs to the subsonic region or the transonic curve. We employ the method of $a$-contraction with shifts to prove that, if both the shock strength and the initial perturbation are suitably small, and the viscous shock is far from the outflow boundary, then the solution asymptotically converges to the viscous shock up to a dynamical shift. We also prove that the speed of the time-dependent shift decays to zero as time goes to infinity, so that the shifted viscous shock still retains its original profile asymptotically. Since the outflow problem in the Lagrangian mass coordinate leads to a free-boundary problem due to the absence of a boundary condition for the fluid density, we consider the outflow problem in the original Eulerian coordinate instead. Although the method of $a$-contraction with shifts is technically more complicated in the Eulerian coordinate than in the Lagrangian one, this provides a more favorable framework by avoiding the difficulties arising from a free boundary. Note that this is the first result on the time-asymptotic stability of viscous shock for the outflow problem of the Navier-Stokes equations.