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Characteristic Boundary Layers for Mixed Hyperbolic‐Parabolic Systems in One Space Dimension and Applications to the Navier‐Stokes and MHD Equations
Author(s) -
Bianchini Stefano,
Spinolo Laura V.
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21892
Subject(s) - mathematics , mathematical analysis , boundary (topology) , magnetohydrodynamics , dimension (graph theory) , space (punctuation) , hyperbolic partial differential equation , eulerian path , viscosity , compressibility , boundary value problem , limit (mathematics) , partial differential equation , physics , mechanics , pure mathematics , quantum mechanics , lagrangian , linguistics , philosophy , plasma
We provide a detailed analysis of the boundary layers for mixed hyperbolic‐parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so‐called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so‐called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic‐parabolic system and for the hyperbolic system obtained by neglecting the second‐order terms. Our analysis applies in particular to the compressible Navier‐Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems. © 2020 Wiley Periodicals, Inc.