The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains. (English) Zbl 1240.35412

Mitrea, Marius; Monniaux, Sylvie

Differ. Integral Equ. 22, No. 3-4, 339-356 (2009).

The authors investigate the Navier-Stokes equations in a three-dimensional bounded Lipschitz domain \(\Omega \), equipped with “free boundary” conditions. Applying the Fujita-Kato method they prove the existence of a local mild solution. The free boundary conditions means that the following boundary condition \(\nu \cdot u=0\) and \(\nu \times \operatorname {curl} u = 0\), where \(\nu \) is the outward unit normal to \(\Omega \) and \(u\) is the velocity field. The approach uses the properties of the Hodge-Laplacian in Lipschitz domain.

Reviewer: Šárka Nečasová (Praha)

MSC:

35Q35	PDEs in connection with fluid mechanics
76D05	Navier-Stokes equations for incompressible viscous fluids

Keywords:

Navier-Stokes equations; local mild solution; Lipschitz domain