$L^q$-Theory of Navier-Stokes Flows in General Unbounded Domains

We report on a joint paper by H. Kozono, H. Sohr and R. Farwig (Acta Math. 195) dealing with the instationary Navier-Stokes equations in a general unbounded domain of $\text{R}3$. It is known by counter-examples that the usual $L^q$-approach to the Stokes equations, well known e.g. for bounded and exterior domains, cannot be extended to general domains $\mathrm{\Omega }\subseteq \text{R}3$ without any modification for $q\ne 2$. However, we will show that important properties like Helmholtz decomposition, analyticity of the Stokes semigroup, and the maximal regularity estimate of the nonstationary Stokes equations remain valid for general unbounded smooth domains even for $q\ne 2$ if we replace the space $L^q$ for $2\le q<\mathrm{\infty }$ by the intersection $L2\cap L^q$ and for $1<q<2$ by the sum space $L2+L^q$. As an application we prove for general $\mathrm{\Omega }$ the existence of a (suitable) weak solution $u$ of the Navier-Stokes equations with pressure term $\mathrm{\nabla }p\in L_{\text{loc}}^{5/4}$, conjectured by Caffarelli-Kohn-Nirenberg (1982), and satisfying both the local and strong energy inequality.