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Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids
Author(s) -
Klainerman Sergiu,
Majda Andrew
Publication year - 1981
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.3160340405
Subject(s) - mathematics , compressibility , limit (mathematics) , compressible flow , mathematical analysis , mach number , nonlinear system , constructive , fluid dynamics , flow (mathematics) , physics , mechanics , geometry , process (computing) , quantum mechanics , computer science , operating system
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.