LAUSR

Abstract This paper studies the Stokes system - Δ ⁢ ???? + ∇ ⁡ ρ = ???? {-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}} , ∇ ⋅ ???? = χ {\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}} on ⁢ ∂ ⁡ Ω , \displaystyle\phantom{}\text{on }\partial\Omega, ???? ⋅ ???? = ???? ⋅ ???? , \displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g}, ???? ⋅ [ ∂ ⁡ ???? ∂ ⁡ ???? - ρ ⁢ ???? + b ⁢ ???? ] = ???? ⋅ τ \displaystyle{\boldsymbol{\tau}}\cdot\bigg{[}\frac{\partial{\mathbf{u}}}{% \partial{\mathbf{n}}}-\rho{\mathbf{n}}+b{\mathbf{u}}\bigg{]}={\mathbf{h}}\cdot\tau on ⁢ ∂ ⁡ Ω , \displaystyle\phantom{}\text{on }\partial\Omega, ???? ⋅ ???? = ???? ⋅ ???? , \displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}, [ T ⁢ ( ???? , ρ ) ⁢ ???? + b ⁢ ???? ] ⋅ τ = ???? ⋅ τ \displaystyle[T({\mathbf{u}},\rho){\mathbf{n}}+b{\mathbf{u}}]\cdot\tau={% \mathbf{h}}\cdot\tau on ⁢ ∂ ⁡ Ω . \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 ⁢ ( Ω ; ℝ 2 ) × W s - 1 , q ⁢ ( Ω ) {W^{s,q}(\Omega;{\mathbb{R}}^{2})\times W^{s-1,q}(\Omega)} , with 1 + 1 / q < s < ∞ {1+1/q