On the stability of approximate solutions of hyperbolic-parabolic systems and the all-time existence of smooth, slightly compressible flows

Abstract

We consider hyperbolic-parabolic systems of nonlinear partial differential equations. While such systems are difficult to analyze in general, physical applications often display interesting singular limits for which strong solutions may be constructed. These are viewed as approximate solutions of the original problem for parameter values near the singular limit. Restricting ourselves to problems posed in all space, we show that if: (i) The full system linearized about a constant state is dissipative in a special inner product; (ii) The approximate solution and the defect resulting from its substitution into the full system satisfy mild decay conditions; then for all initial data sufficiently close to the approximate solution, a classical solution of the full system exists for all time which remains close to it. We also show how the inner product required in (i) can be constructed when the linearized problem involves a large, symmetric, hyperbolic part and when the coupling between the hyperbolic and parabolic subsystems is nonvanishing. This enables us to apply the general result to the equations of compressible, heat-conducting fluids at low Mach number. In this case the approximate solution is simply determined by a solution of the incompressible Navier-Stokes equations.