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

TLDR

Classical physics problems often involve studying the limiting behavior of quasilinear hyperbolic systems as coefficients become infinite. The authors aim to develop a general theory for such singular limit problems using classical methods. The theory relies on balancing nonlinear structural conditions on system matrix coefficients with suitable initialization procedures. The theory enables analysis of singular limits in compressible fluid flow and magneto‑fluid dynamics, providing constructive local existence theorems, a self‑contained proof of convergence to incompressible limits as the Mach number shrinks, and analogous results for Navier–Stokes equations with vanishing viscosity.

Abstract

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.