Nonlinear stability of planar viscous shock wave to three-dimensional compressible Navier-Stokes equations

Abstract

We prove the nonlinear stability of the planar viscous shock up to a time-dependent shift for the three-dimensional (3D) compressible Navier-Stokes equations under the generic perturbations, in particular, without zero mass conditions. Moreover, the time-dependent shift function keeps the shock profile shape time-asymptotically. Our stability result is unconditional for the weak planar Navier-Stokes shock. Our proof is motivated by the $a$-contraction method (a kind of weighted $L^2$-relative entropy method) with time-dependent shift introduced in [10,11,13] for the stability of viscous shock in one-dimensional (1D) case. Instead of the classical anti-derivative techniques, we perform the stability analysis of planar Navier-Stokes shock in original $H^2$-perturbation framework and therefore zero mass conditions are not necessarily needed, which, in turn, brings out the essential difficulties due to the compressibility of viscous shock. Furthermore, compared with 1D case, there are additional difficulties coming from the wave propagation along the multi-dimensional transverse directions and their interactions with the viscous shock. To overcome these difficulties, a multi-dimensional version sharp weighted Poincar${\rm \acute{e}}$ inequality (see Lemma 3.1), $a$-contraction techniques with time-dependent shift, and some essential physical structures of the multi-dimensional Navier-Stokes system are fully used.