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

Briane, Marc; Casado-Díaz, Juan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1066-1087.

In this paper, a $W^{-1,N^{\text{'}}}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on ${ℝ}^N$, or on a regular bounded open set of ${ℝ}^N$. 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.

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