THE STATIONARY NAVIER-STOKES
SYSTEM IN NONSMOOTH MANIFOLDS
Martin Dindoš∗ (jointly with Marius Mitrea)
The Navier-Stokes equation is one of the most studied nonlinear equation.
It models the flow of an incompressible viscous fluid.
Due to nonlinear nature of these equations, one ingredient in their analysis
is the study of stationary linearized systems. Traditionally, the linearized
system known as the Stokes system is the primary object of such analysis.
In our work we introduce and study whole family of such linearized systems
among which the Stokes system is only one of many others.
We find the solution to such linear system by the method of layer po-
tentials. The whole approach works in very general framework of Lipschitz
domains on Riemannian manifolds. We are particulary interested in what
range of Sobolev-Besov spaces the solution exists. Our results show that the
smoothness of the solution depends on the smoothness of the boundary, that
is for domains with C 1 boundary or boundary with small Lipschitz constant
we get solvability in full range of spaces. In general Lipschitz domain this
range becomes restricted.
Next we apply the Schauder fixed point theorem to obtain the existence
result for the stationary Navier-Stokes equation. Our result shows existence
for arbitrary large data in up to four dimensions and small data in higher
dimensions.
We will also present several open problems as well as suggest possible
ways to solve them.
∗
Department of Mathematics, Brown University, Providence RI
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Converted from dindos.pdf (PDF).