The initial-value problem for the Navier-Stokes equations with a free surface in \(L^q\)-Sobolev spaces. (English) Zbl 1105.35072

Abels, Helmut

Adv. Differ. Equ. 10, No. 1, 45-64 (2005).

The author deals with the time-dependent flow of an incompressible, viscous fluid bounded above by a free surface and below by a fixed bottom under the influence of gravity. The author shows short-time existence of strong solutions of this free-boundary problem in the setting of anisotropic \(L^q\)-Sobolev spaces for \(q> n\) in space dimension \(n\geq 2\).

Reviewer: Messoud A. Efendiev (Berlin)

MSC:

35Q30	Navier-Stokes equations
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07	Stokes and related (Oseen, etc.) flows
76D27	Other free boundary flows; Hele-Shaw flows

Keywords:

free surface; anisotropic Sobolev space; strong solution; short-time existence