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Estimate of the pressure when its gradient is the divergence of a measure. Applications

Published online by Cambridge University Press:  28 October 2010

Marc Briane  and
Juan Casado-Díaz
Marc Briane
Affiliation:
Institut de Recherche Mathématique de Rennes, INSA de Rennes, France. mbriane@insa-rennes.fr
Juan Casado-Díaz
Affiliation:
Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain. jcasadod@us.es
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Abstract

In this paper, a $W^{-1,N'}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on $\mathbb R^N$, or on a regular bounded open set of $\mathbb R^N$. The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math.23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc.9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an existence result for the stationary Navier-Stokes equation when the viscosity tensor is only in L1.

Keywords

PressureNavier-Stokes equationdiv-curlmeasure datafundamental solution
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1066 - 1087
Copyright
© EDP Sciences, SMAI, 2010

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