Maximal [formula omitted]- [formula omitted] regularity of the linearized initial–boundary value problem for motion of compressible viscous fluids

Abstract

We discuss the initial–boundary value problem for the linearized system of equations which describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in the anisotropic Sobolev space W q 1 ( W ˙ p 1 ) × ( W p , q 2 , 1 ) n for any 1 < p < ∞ , 1 < q < ∞ globally in time. Moreover, exponentially weighted L p - L q estimates for solutions globally in time can be established. We prove the above properties by resolvent estimates for the linearized operator of the above system, the theory of analytic semigroups on Banach spaces and the operator-valued Fourier multiplier theorem on UMD spaces.