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- W4285777416 abstract "The Stokes resolvent problem $lambda u - Delta u + nabla phi = f$ with $mathrm{div}(u) = 0$ 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 $mathrm{L}^2_{sigma} (Omega) ni f mapsto pi in mathrm{L}^2 (Omega)$ decays like $lvert lambda rvert^{- 1 / 2}$ which agrees exactly with the scaling of the equation. In comparison to that, we show that the operator norm of this mapping under Dirichlet boundary conditions decays like $lvert lambda rvert^{- alpha}$ for $0 leq alpha < 1 / 4$ and we show that this decay rate cannot be improved to any exponent $alpha > 1 / 4$, 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. 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 on general Lipschitz domains and is to the best knowledge of the author the first result that provides $mathrm{L}^p$-estimates for the Stokes resolvent subject to Neumann-type boundary conditions on general convex domains." @default.
- W4285777416 created "2022-07-19" @default.
- W4285777416 creator A5053509485 @default.
- W4285777416 date "2019-11-14" @default.
- W4285777416 modified "2023-10-14" @default.
- W4285777416 title "The Stokes resolvent problem: Optimal pressure estimates and remarks on resolvent estimates in convex domains" @default.
- W4285777416 doi "https://doi.org/10.48550/arxiv.1911.06231" @default.
- W4285777416 hasPublicationYear "2019" @default.
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