Sharp existence results for the stationary Navier-Stokes problem in three-dimensional exterior domains
The stationary Navier-Stokes system is considered in an exterior domain \(\Omega\subset \mathbb{R}^3\) \[ \begin{aligned} -&\Delta v+\lambda v\cdot\nabla v+\nabla p=f,\quad \nabla\cdot v=0 \quad \text{in \;} \Omega,\\&v=v_0 \quad \text{on \;}\partial \Omega,\quad \lim_{|x|\to\infty}v(x)= v_{\infty}.\end{aligned} \] The authors investigate this problem in specific Sobolev-like spaces. Using the theory of Fredholm operators and topological degree arguments they establish the existence of solutions and the finiteness of the number of solutions for fixed \(v_0\) and generic \(f\). In particular the existence is obtained for an arbitrary \(f\in L^{\frac 43}(\Omega)\).
scientific article; zbMATH DE number 1538595