Resolvent estimates for the linearized operator in L p associated with motion of compressible viscous fluids

Abstract

We consider the resolvent problem for the linearized system of equations that describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in $${\dot{W}^{1}_{p} \times (W^{2}_{p})^{n}}$$ satisfying L p estimates for any 1 < p < ∞. Moreover, resolvent estimates for the linearized operator of the above system in $${\dot{W}^{1}_{p} \times (L_{p})^{n}}$$ are established. Our main results yield clearly that the linearized operator is the infinitesimal generator of a uniformly bounded analytic semigroup on $${\dot{W}^{1}_{p} \times (L_{p})^{n}}$$ .