Several non-standard problems for the stationary Stokes system
Abstract This paper studies the Stokes system - Δ ⁢ 𝐮 + ∇ ⁡ ρ = 𝐟 {-\Delta{\mathbf{u}}+\nabla\rho={\mathbf{f}}} , ∇ ⋅ 𝐮 = χ {\nabla\cdot{\mathbf{u}}=\chi} in Ω with three boundary conditions: 𝐧 ⋅ 𝐮 = 𝐧 ⋅ 𝐠 , \displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}, 𝐧 × ( ∇ × 𝐮 ) = 𝐧 × 𝐡 \displaystyle{\mathbf{n}}\times(\nabla\times{\mathbf{u}})={\mathbf{n}}\times{% \mathbf{h}} on ⁢ ∂ ⁡ Ω , \displaystyle\phantom{}\text{on }\partial\Omega, 𝐧 ⋅ 𝐮 = 𝐧 ⋅ 𝐠 , \displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={\mathbf{n}}\cdot\mathbf{g}, 𝝉 ⋅ [ ∂ ⁡ 𝐮 ∂ ⁡ 𝐧 - ρ ⁢ 𝐧 + b ⁢ 𝐮 ] = 𝐡 ⋅ 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, 𝐧 ⋅ 𝐮 = 𝐧 ⋅ 𝐠 , \displaystyle{\mathbf{n}}\cdot{\mathbf{u}}={{\mathbf{n}}\cdot\mathbf{g}}, [ T ⁢ ( 𝐮 , ρ ) ⁢ 𝐧 + b ⁢ 𝐮 ] ⋅ τ = 𝐡 ⋅ 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<s<\infty} , in Besov spaces B s q , r ⁢ ( Ω ; ℝ 2 ) × B s - 1 q , r ⁢ ( Ω ) {B_{s}^{q,r}(\Omega;{\mathbb{R}}^{2})\times B_{s-1}^{q,r}(\Omega)} , with 1 + 1 / q < s < ∞ {1+1/q<s<\infty} , and classical solutions in 𝒞 k , α ⁢ ( Ω ¯ , ℝ 2 ) × 𝒞 k - 1 , α ⁢ ( Ω ¯ ) {{\mathcal{C}}^{k,\alpha}(\overline{\Omega},{\mathbb{R}}^{2})\times{\mathcal{C% }}^{k-1,\alpha}(\overline{\Omega})} , with 0 < α < 1 {0<\alpha<1} , k ∈ ℕ {k\in{\mathbb{N}}} .