## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 1 (2005), 45-64.

### The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces

#### Abstract

We prove small-time existence of strong solutions of a free-boundary-value problem, which
describes the motion of an incompressible viscous fluid occupying a semi-infinite domain
bounded above by a free surface. This problem was studied by Beale [6] and others in
$L^2$-Sobolev spaces. In contrast to the latter contribution we study solutions in
$L^q$-Sobolev spaces for $q>n$ in space dimension $n\geq 2$. This approach has the
advantage that the regularity assumptions can be reduced in comparison to [6]. In order to
solve the linearized system, we use the nonstationary *reduced* Stokes equations with
a mixed boundary condition and the maximal regularity of the associated reduced Stokes
operator.

#### Article information

**Source**

Adv. Differential Equations Volume 10, Number 1 (2005), 45-64.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867895

**Mathematical Reviews number (MathSciNet)**

MR2106120

**Zentralblatt MATH identifier**

1105.35072

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 35R35: Free boundary problems 76D03: Existence, uniqueness, and regularity theory [See also 35Q30] 76D05: Navier-Stokes equations [See also 35Q30]

#### Citation

Abels, Helmut. The initial-value problem for the Navier-Stokes equations with a free surface in $L^q$-Sobolev spaces. Adv. Differential Equations 10 (2005), no. 1, 45--64. https://projecteuclid.org/euclid.ade/1355867895.