The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains
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Calculus of Variations
The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains Patrick Tolksdorf1 Received: 18 November 2019 / Accepted: 30 June 2020 © The Author(s) 2020
Abstract The Stokes resolvent problem λu − u + ∇φ = f with 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 Lσ2 () f → φ ∈ L2 () decays like |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 for 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 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 (; Cd ) admit H2 -regularity and further prove localized H2 -estimates for the Stokes resolvent problem. By a generalized version of Shen’s L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in L p (; Cd ) for 2d/(d + 2) < p < 2d/(d − 2) (with 1 < p < ∞ if 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. Mathematics Subject Classification 35Q35 · 47A10 · 42B37
Communicated by Y. Giga. The author was supported by the Project ANR INFAMIE (ANR-15-CE40-0011).
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Patrick Tolksdorf [email protected] Institut für Mathematik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany 0123456789().: V,-vol
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P. Tolksdorf
1 Introduction The main object under investigation is the Stokes resolvent problem in a bounded domain ⊂ Rd λu − u + ∇φ = f in (Res) div(u) = 0 in . The resolvent parameter λ is supposed to be contained in a sector Sθ , θ ∈ [0, π), in the complex plane, i.e., Sθ := {z ∈ C \ {0} : |arg(z)| < θ } if θ ∈ (0, π) and S0 := (0, ∞). In this article, this system is complemented with two different types of boundary conditions. There are the homogeneous Dirichlet boundary conditions u=0
on ∂
(Dir)
and a family of homogeneous Neumann-type boundary conditions which read {Du + μ[Du] }n − φn = 0
on ∂.
(Neu)
Here μ ∈ (−1, 1] is a parameter, n denotes the outward unit normal to , and Du the Jacobi-matrix of u. There is a tremendous literature on these equations on different types of domains, see, e.g., [1,6,7,16,17,23,26,35,42,43,50,56,58] to mention only a few. Notice that the Neumann-type boundary condition with μ = 1 plays an eminent role in
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