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Local Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions

Local Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions We study the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions. It is known that if the Dirichlet boundary conditions are prescribed on thewhole boundary and the total influx equals zero, then weak solutions exist globally in time andthey are even unique and smooth in the case of two-dimensional domains. The methods that havebeen used to prove these results fail if non-Dirichlet conditions are applied on a part of theboundary, since there is then no control over the energy flux on this part of the boundary. In thispaper, we prove the existence and the uniqueness of solutions on a (short) time interval. Theproof is performed for Lipschitz domains and a wide class of initial data. The length of the timeinterval on which the solution exists depends only on certain norms of the data. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Local Solutions to the Navier–Stokes Equations with Mixed Boundary Conditions

Acta Applicandae Mathematicae , Volume 54 (3) – Sep 18, 2004

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References (11)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Kluwer Academic Publishers
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1023/A:1006185601807
Publisher site
See Article on Publisher Site

Abstract

We study the existence and uniqueness of solutions to the system of three-dimensionalNavier-Stokes equations and the continuity equation for incompressible fluid with mixedboundary conditions. It is known that if the Dirichlet boundary conditions are prescribed on thewhole boundary and the total influx equals zero, then weak solutions exist globally in time andthey are even unique and smooth in the case of two-dimensional domains. The methods that havebeen used to prove these results fail if non-Dirichlet conditions are applied on a part of theboundary, since there is then no control over the energy flux on this part of the boundary. In thispaper, we prove the existence and the uniqueness of solutions on a (short) time interval. Theproof is performed for Lipschitz domains and a wide class of initial data. The length of the timeinterval on which the solution exists depends only on certain norms of the data.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Sep 18, 2004

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