Abstract
The Stokes resolvent problem lambda u - Delta u + nabla phi = f with {text {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 mathrm {L}^2_{sigma } (Omega ) ni f mapsto phi in mathrm {L}^2 (Omega ) decays like |lambda |^{- 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 |lambda |^{- alpha } for 0 le alpha le 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 Omega 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 in mathrm {L}^2 (Omega ; {mathbb {C}}^d) admit mathrm {H}^2-regularity and further prove localized mathrm {H}^2-estimates for the Stokes resolvent problem. By a generalized version of Shen’s mathrm {L}^p-extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in mathrm {L}^p (Omega ; {mathbb {C}}^d) for 2d / (d + 2)< p < 2d / (d - 2) (with 1< p < infty 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.
Highlights
The main object under investigation is the Stokes resolvent problem in a bounded domain ⊂ Rd λu − u + ∇φ = f in div(u) = 0 in . (Res)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, ∞)
We investigate two different questions: Question 1: The first question deals with the behavior of the operator norm of the mapping f → φ with respect to λ, i.e., we seek an inequality of the form φ L2( ) ≤ C (λ) f L2( ;Cd ) ( f ∈ Lσ2 ( ))
Let f ∈ C∞ σ ( k), let u be given by u := (λ + Bμ,k )−1 f, and let φ denote the associated pressure
Summary
We give first results in this direction for the Stokes resolvent problem (Res) subject to the Neumann-ty√pe boundary condition (Neu), we restrict the interval of parameters μ to be (−1, 2 − 1). This still includes the case μ = 0 the physically important case μ = 1 is excluded. For some constant C > 0 depending only on d, θ , and μ, see Theorem 4.4 This implies that solutions u and φ to − u + ∇φ = f and div(u) = 0 for some f ∈ Lσ2 ( ) and subject to the boundary condition (Neu) satisfy u ∈ H2( ; Cd ) and φ ∈ H1( ). This section is devoted to present results concerning the Stokes resolvent problem (Res) subject to no-slip boundary conditions (Dir) and subject to Neumann-type boundary conditions (Neu)
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