How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax
  Remote Access

Shock waves in conservation laws with physical viscosity


About this Title

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

View full volume PDF

View other years and numbers:

Table of Contents


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.

References [Enhancements On Off] (What's this?)