Weak Neumann implies Stokes (Q2910995)

Let \(\Omega\) be a subset of \({\mathbb{R}}^n\) with a uniformly \(C^3\)-boundary (not necessarily compact). Consider the Stokes equation \(u_t - \Delta u + \nabla \pi = f, \;\mathrm{div}\, u = 0 \text{ in } \Omega\times (0,T); \;u=0 \text{ on } \partial \Omega \times (0,T); \;u(0)=u_0 \text{ in } \Omega\). Assume that the Helmoltz projection on \(L^p(\Omega )\) exists for some \(1<p< \infty\). The authors show that the corresponding Stokes operator generates an analytic semigroup on \(L^p_{\sigma}(\Omega )\) and that the solution of the Stokes equation satisfies the maximal \(L^q\)-\(L^p\)-regularity estimate. As a second main result they obtain the existence of a local mild solution of the Navier-Stokes equations defined on such sets \(\Omega\) provided \(p>n\).