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

Marc Briane (1) , Juan Casado-Diaz (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 EDAN US - Departamento de Ecuaciones Diferenciales y Analisis Numerico

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 L 1.

Mots clés: Pressure; Navier-Stokes equation; div-curl; measure data; fundamental solution

Domaines: Analyse numérique [math.NA]

Marc Briane, Juan Casado-Diaz. Estimate of the pressure when its gradient is the divergence of a measure. Applications. ESAIM: Control, Optimisation and Calculus of Variations, 2011, 17 (4), pp.1066-1087. ⟨10.1051/cocv/2010037⟩. ⟨hal-00567605⟩