Special Session 27: Recent Trends in Navier-Stokes Equations, Euler Equations, and Related Problems
On the Helmholtz decomposition in general domains
Werner Varnhorn
Kassel University
Germany
Co-Author(s):    Reinhard Farwig, Christian Simader, Hermann Sohr
Abstract:
In the theory of the incompressible Navier-Stokes equatuons, the Helmholtz decomposition (HD) in $L^q(\varOmega)$ plays a fundamental role. We show for general domains $\varOmega\subseteq \mathbb R^n$, $n\ge 2$, $1 < q < \infty$ that (HD) is necessary and sufficient for the validity of a certain gradient estimate (GE), and also for the validity of a certain estimate (DE) for divergence free functions. Moreover, we study the optimal constants in the estimates (GE) and (DE) and prove that these constants coincide.
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