Shock waves in conservation laws with physical viscosity

We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to...

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Bibliographic Details
Main Authors: Liu, Tai-Ping, Zeng, Yanni
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2015
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Conservation laws (Mathematics)
Green''s functions
Mathematics
Shock waves
Shock waves > Mathematics
Conservation laws
Green’s function
nonlinear stability
wave interactions
compressible Navier-Stokes equations
magneto-hydrodynamics
physical viscosity
large time behavior
pointwise estimates
quasilinear hyperbolic-parabolic systems
shock waves
ISBN:9781470410162, 1470410168
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Summary:We study the perturbation of a shock wave in conservation laws with physical viscosity. We obtain the detailed pointwise estimates of the solutions. In particular, we show that the solution converges to a translated shock profile. The strength of the perturbation and that of the shock are assumed to be small, but independent. Our assumptions on the viscosity matrix are general so that our results apply to the Navier-Stokes equations for the compressible fluid and the full system of magnetohydrodynamics, including the cases of multiple eigenvalues in the transversal fields, as long as the shock is classical. Our analysis depends on accurate construction of an approximate Green’s function. The form of the ansatz for the perturbation is carefully constructed and is sufficiently tight so that we can close the nonlinear term through the Duhamel’s principle.
Bibliography:Includes bibliographical references (p. 167-168)
Volume 234, number 1105 (fifth of 5 numbers), March 2015
ISBN:9781470410162
1470410168
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1105