Navier–Stokes regularization of multidimensional Euler shocks

Abstract

We establish existence and stability of multidimensional shock fronts in the vanishing viscosity limit for a general class of conservation laws with “real”, or partially parabolic, viscosity including the Navier–Stokes equations of compressible gas dynamics with standard or van der Waals-type equation of state. More precisely, given a curved Lax shock solution u 0 of the corresponding inviscid equations for which (i) each of the associated planar shocks tangent to the shock front possesses a smooth viscous profile and (ii) each of these viscous profiles satisfies a uniform spectral stability condition expressed in terms of an Evans function, we construct nearby smooth viscous shock solutions u ε of the viscous equations converging to u 0 as viscosity ε → 0 , and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i)–(ii) hold always for shock waves of sufficiently small amplitude, but in general may fail for large amplitudes. We treat the viscous shock problem considered here as a representative of a larger class of multidimensional boundary problems arising in the study of viscous fluids, characterized by sharp spectral conditions rather than symmetry hypotheses, which can be analyzed by Kreiss-type symmetrizers. Compared to the strictly parabolic (artificial viscosity) case, the main new features of the analysis appear in the high frequency estimates for the linearized problem. In that regime we use frequency-dependent conjugators to decouple parabolic components that are smoothed from hyperbolic components (like density in Navier–Stokes) that are not. The construction of the conjugators and the subsequent estimates depend on a careful spectral analysis of the linearized operator.