L^p‐estimates of the Stokes resolvent with nonhomogeneous Dirichlet boundary conditions in 3D exterior domains

Abstract

The article deals with the homogeneous Stokes resolvent system in a 3D exterior domain, under inhomogeneous Dirichet boundary conditions. Solutions to this boundary value problem are estimated in Lp‐norms, with the bounds in these estimates depending on the absolute value of the resolvent parameter λ in an explicit way. Two types of boundary data are considered, that is, Lp‐ and W2 − 1/p, p‐data. It is shown in particular that the Lp‐norm of the velocity outside a vicinity of the boundary, after subtraction of the gradient of a certain harmonic function, is bounded by a constant times |λ|−1 ‖b‖p. This estimate carries over to the Oseen resolvent system, leading to a result which has applications in the theory of spatial asymptotics of solutions to the 3D time‐dependent Navier‐Stokes system with Oseen term.

Time-dependent incompressible viscous flows around a rigid body: Estimates of spatial decay independent of boundary conditions

“…where n (A) : ∂Ω → R 3 denotes the outward unit normal to A. The next theorem, which the title of this section alludes to, and which we take from [24], deals with the Oseen resolvent problem…”