The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains

Abstract

We investigate the Navier-Stokes equations in a suitable functional setting, in a three-dimensional bounded Lipschitz domain $\Omega$, equipped with ``free boundary'' conditions. In this context, we employ the Fujita-Kato method and prove the existence of a local mild solution. Our approach makes essential use of the properties of the Hodge-Laplacian in Lipschitz domains.