Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Sergiu Klainerman, Andrew Majda

Research output: Contribution to journalArticle

Abstract

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto‐fluid dynamics including new constructive local existence theorems for the time‐singular limit equations. In particular, the authors give an entirely self‐contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier‐Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.

Original languageEnglish (US)
Pages (from-to)481-524
Number of pages44
JournalCommunications on Pure and Applied Mathematics
Volume34
Issue number4
DOIs
StatePublished - 1981

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Incompressible Limit
Quasilinear Hyperbolic System
Singular Limit
Compressible Fluid
Mach number
Flow of fluids
Physics
Viscosity
Fluids
Coefficient
Incompressible Navier-Stokes
Convergence of Solutions
Local Existence
Limiting Behavior
Compressible Flow
Initialization
Existence Theorem
Fluid Flow
Navier-Stokes Equations
Tend

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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