The Stokes resolvent problem : optimal pressure estimates and remarks on resolvent estimates in convex domains

Abstract

The Stokes resolvent problem 𝜆𝑢−Δ𝑢+∇𝜙=𝑓 λ u − Δ u + ∇ ϕ = f with div(𝑢)=0 div ( u ) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of L2𝜎(Ω)∋𝑓↦𝜙∈L2(Ω) L σ 2 ( Ω ) ∋ f ↦ ϕ ∈ L 2 ( Ω ) decays like |𝜆|−1/2 | λ | − 1 / 2 which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like |𝜆|−𝛼 | λ | − α for 0≤𝛼≤1/4 0 ≤ α ≤ 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain Ω Ω is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side 𝑓∈L2(Ω;ℂ𝑑) f ∈ L 2 ( Ω ; C d ) admit H2 H 2 -regularity and further prove localized H2 H 2 -estimates for the Stokes resolvent problem. By a generalized version of Shen’s L𝑝 L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in L𝑝(Ω;ℂ𝑑) L p ( Ω ; C d ) for 2𝑑/(𝑑+2)<𝑝<2𝑑/(𝑑−2) 2 d / ( d + 2 ) < p < 2 d / ( d − 2 ) (with 1<𝑝<∞ 1 < p < ∞ if 𝑑=2 d = 2 ). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.