‐solution of the Neumann, Robin and transmission problem for the scalar Oseen equation

Medková, Dagmar

doi:10.1002/mana.201600307

Mathematische Nachrichten

2017

DOI: 10.1002/mana.201600307

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‐solution of the Neumann, Robin and transmission problem for the scalar Oseen equation

Dagmar Medková

Abstract: We find necessary and sufficient conditions for the existence of an Lq‐solution of the Neumann problem, the Robin problem and the transmission problem for the scalar Oseen equation in three‐dimensional open sets. As a consequence we study solutions of the generalized jump problem.

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2020

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Several non-standard problems for the stationary Stokes system

Medková

2020

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This paper studies the Stokes system {-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}}, {\nabla\cdot{\mathbf{u}}=\chi} in Ω with three boundary conditions:\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}},\displaystyle{\mathbf{n}}\times(\nabla\times{\mathbf{u}})={\mathbf{n}}\times{% \mathbf{h}}\displaystyle\phantom{}\text{on }\partial\Omega,\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g},\displaystyle{\boldsymbol{\tau}}\cdot\bigg{[}\frac{\partial{\mathbf{u}}}{% \partial{\mathbf{n}}}-\rho{\mathbf{n}}+b{\mathbf{u}}\bigg{]}={\mathbf{h}}\cdot\tau\displaystyle\phantom{}\text{on }\partial\Omega,\displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}},\displaystyle[T({\mathbf{u}},\rho){\mathbf{n}}+b{\mathbf{u}}]\cdot\tau={% \mathbf{h}}\cdot\tau\displaystyle\phantom{}\text{on }\partial\Omega.Here Ω is a bounded simply connected planar domain. We find a necessary and sufficient condition for the existence of a solution in Sobolev spaces {W^{s,q}(\Omega;{\mathbb{R}}^{2})\times W^{s-1,q}(\Omega)}, with {1+1/q<s<\infty}, in Besov spaces {B_{s}^{q,r}(\Omega;{\mathbb{R}}^{2})\times B_{s-1}^{q,r}(\Omega)}, with {1+1/q<s<\infty}, and classical solutions in {{\mathcal{C}}^{k,\alpha}(\overline{\Omega},{\mathbb{R}}^{2})\times{\mathcal{C% }}^{k-1,\alpha}(\overline{\Omega})}, with {0<\alpha<1}, {k\in{\mathbb{N}}}.

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Several non-standard problems for the stationary Stokes system

Medková

2020

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‐solution of the Neumann, Robin and transmission problem for the scalar Oseen equation

Abstract: We find necessary and sufficient conditions for the existence of an Lq‐solution of the Neumann problem, the Robin problem and the transmission problem for the scalar Oseen equation in three‐dimensional open sets. As a consequence we study solutions of the generalized jump problem.

Cited by 1 publication

References 46 publications

Several non-standard problems for the stationary Stokes system

Several non-standard problems for the stationary Stokes system

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