L p ‐estimates of the Stokes resolvent with nonhomogeneous Dirichlet boundary conditions in 3D exterior domains
Paul Deuring
Source 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 L p ‐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, L p ‐ and W 2 − 1/ p , p ‐data. It is shown in particular that the L p ‐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.
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