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On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface

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Zajączkowski W. M.

Instytut Matematyczny Polskiej Akademii Nauk

We consider the motion of a viscous compressible barotropic fluid in $ℝ^3$ bounded by a free surface which is under constant exterior pressure. For a given initial density, initial domain and initial velocity we prove the existence of local-in-time highly regular solutions. Next assuming that the initial density is sufficiently close to a constant, the initial pressure is sufficiently close to the external pressure, the initial velocity is sufficiently small and the external force vanishes we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

CONTENTS 1. Introduction.......................................5 2. Global estimates and relations........11 3. Local existence...............................16 4. Global differential inequality............44 5. Korn inequality................................81 6. Global existence.............................89 References.......................................100

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1 Zajączkowski W. M.

1 1993

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On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface