Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations

Authors:
H.-O Kreiss;J Lorenz;M.J Naughton
Affiliations:
Department of Mathematics, UCLA, Los Angeles, California 90024 U.S.A.;Applied Mathematics, Caltech, Pasadena, California 91125 U.S.A.;Meterology Research Center, Melbourne Victoria 3001, Australia
Venue:
Advances in Applied Mathematics
Year:
1991

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Abstract

We study the slightly compressible Navier-Stokes equations. We first consider the Cauchy problem, periodic in space. Under appropriate assumptions on the initial data, the solution of the compressible equations consists-to first order-of a solution of the incompressible equations plus a function which is highly oscillatory in time. We show that the highly oscillatory part (the sound waves) can be described by wave equations, at least locally in time. We also show that the bounded derivative principle is valid; i.e., the highly oscillatory part can be suppressed by initialization. Besides the Cauchy problem, we also consider an initial-boundary value problem. At the inflow boundary, the viscous term in the Navier-Stokes equations is important. We consider the case where the compressible pressure is prescribed at inflow. In general, one obtains a boundary layer in the pressure; in the velocities a boundary layer is not present to first approximation.