Several non-standard problems for the stationary Stokes system

Abstract

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}}}.

Least-squares formulations for Stokes equations with non-standard boundary conditions -- A unified approach

“…We plan to discuss exponentially accurate least-squares spectral formulations for Stokes equations with each of these boundary conditions. 32], [33]) 32], [33])…”

“…These results help us in formulating numerical schemes in appropriate functional spaces. Medkova [32] has studied Stokes equations with three different boundary conditions (B2), (B3), (B4) named as of Navier type on two dimensional bounded domains with Lipschitz boundary. Necessary and sufficient conditions for the existence of solutions on planar domains in Sobolev spaces and Besov spaces have been obtained.…”

Least-squares formulations for Stokes equations with non-standard boundary conditions - A unified approach

“…We plan to discuss exponentially accurate least-squares spectral element formulations for Stokes equations with each of these boundary conditions. 40], [41])…”

“…These results help us in formulating numerical schemes in appropriate functional spaces. Medkova [40] has studied Stokes equations with three different boundary conditions (B2), (B3), (B4) named as of Navier type on two dimensional bounded domains with Lipschitz boundary. Necessary and sufficient conditions for the existence of solutions on planar domains in Sobolev spaces and Besov spaces have been obtained.…”

$L^p$-theory for the exterior Stokes problem with Navier's type slip-without-friction boundary conditions

“…Stokes problem with the boundary conditions (1.3) was studied by many authors. In the one hand, in the bounded domains, one can refer for instance to [26,8,13,15,16]. In the other hand, the case for the exterior domains, we can just mention [4,25].…”