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# memo_has_moved_text();Shock waves in conservation laws with physical viscosity

Tai-Ping Liu and Yanni Zeng

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 234, Number 1105
ISBNs: 978-1-4704-1016-2 (print); 978-1-4704-2032-1 (online)
DOI: http://dx.doi.org/10.1090/memo/1105
Published electronically: August 25, 2014
Keywords:Conservation laws, physical viscosity, shock waves, nonlinear stability, large time behavior, wave interactions, pointwise estimates, Green’s function, compressible Navier-Stokes equations, magneto-hydrodynamics

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Green’s functions for Systems with Constant Coefficients
• Chapter 4. Green’s Function for Systems Linearized Along Shock Profiles
• Chapter 5. Estimates on Green’s Function
• Chapter 6. Estimates on Crossing of Initial Layer
• Chapter 7. Estimates on Truncation Error
• Chapter 8. Energy Type Estimates
• Chapter 9. Wave Interaction
• Chapter 10. Stability Analysis
• Chapter 11. Application to Magnetohydrodynamics

### Abstract

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.