Navier's slip and incompressible limits in domains with variable bottoms
The authors consider heat conducting compressible viscous Stokesian fluids in a domain with a flat top and a variable bottom. The actual rough boundary is approximated by a smooth one by imposing a slip boundary condition on the tangential component of the velocity vector whose normal component vanishes on the smooth boundary. The field equations and boundary conditions involve a small parameter related to the Mach number. Heat conduction is governed by Fourier's law. Shear and bulk viscosity and heat conduction coefficients are assumed to be Lipschitz function of the temperature. It is further supposed that viscous dissipation gives rise to a heat flux at the boundary. The governing differential equations are transformed into integral forms that allow the introduction of weak solutions. By employing various properties of certain Lebesgue and Sobolev spaces, it has been shown that the weak solution of the system in question converges asymptotically to the weak solution corresponding to an incompressible fluid flow with no-slip boundary condition for vanishingly small Mach numbers. Acoustical approximation is also dealt with details.
scientific article; zbMATH DE number 5359512