THE ASYMPTOTIC ANALYSIS OF THE COMPLETE FLUID SYSTEM ON A VARYING DOMAIN: FROM THE COMPRESSIBLE TO THE INCOMPRESSIBLE FLOW

We will present the asymptotic analysis of solutions to the compressible Navier-Stokes Fourier system, when the Mach number is small proportional to epsilon a Froude number is proportional to root epsilon and epsilon -> 0 , and the domain containing the fluid varies with changing parameter e. In particular, the fluid is driven by a gravitation generated by objects placed in the fluid of diameter converging to zero. As epsilon -> 0, we will show that the fluid velocity converges to a solenoidal vector field satisfying the Oberbeck-Boussinesq approximation on R-3 space with a possibly singular gravitational force. Our approach is based on weak solutions. In order to pass to the limit in a convective term we apply the spectral analysis of the associated wave propagator (Neumann Laplacian) governing the motion of acoustic waves.