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Navier-Stokes regularization of multidimensional Euler shocks. (English) Zbl 1173.35082

Summary: 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 \(u^{\varepsilon }\) of the viscous equations converging to \(u^0\) as viscosity \(\varepsilon \rightarrow 0\), and establish for these sharp linearized stability estimates generalizing those of Majda in the inviscid case. Conditions (i) and (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.

MSC:

35L67 Shocks and singularities for hyperbolic equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76L05 Shock waves and blast waves in fluid mechanics
35Q30 Navier-Stokes equations
35L65 Hyperbolic conservation laws
76D33 Waves for incompressible viscous fluids
35B25 Singular perturbations in context of PDEs
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