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ESAIM: Control, Optimisation and Calculus of Variations (1)

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Estimate of the pressure when its gradientis the divergence of a measure. Applications
Marc Briane, Juan Casado-Díaz
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 17 / Issue 4 / October 2011

Published online by Cambridge University Press:

28 October 2010, pp. 1066-1087

Print publication:

October 2011
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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.