Several non-standard problems for the stationary Stokes system

Abstract

AbstractThis 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}}}.