The Two-Dimensional Exterior Stokes Problem, Existence, Regularity and Decay Properties

Abstract

The Stokes problem −Δu+∇p = f, div u = g in Ω, u∣∂Ω = h is investigated for two‐dimensional exterior domains Ω. By means of potential theory, existence, uniqueness and regularity results for weak solutions are proved in weighted Sobolev spaces with weights proportional to ∣x∣δ as ∣x∣→∞. For f = 0,g = 0, explicit decay formulas are obtained for the solutions u and p. Finally, the results are compared with the theory of r‐generalized solutions, i.e. ∇u∈Lr.