Advances in Differential Equations

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

Helmut Abels

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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

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

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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.

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