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

Marc Briane; Juan Casado-Díaz

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1066-1087
  • ISSN: 1292-8119

Abstract

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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 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.

How to cite

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Briane, Marc, and Casado-Díaz, Juan. "Estimate of the pressure when its gradient is the divergence of a measure. Applications." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1066-1087. <http://eudml.org/doc/272952>.

@article{Briane2011,
abstract = {In this paper, a $W^\{-1,N^\{\prime \}\}$ 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.},
author = {Briane, Marc, Casado-Díaz, Juan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {pressure; Navier-Stokes equation; div-curl; measure data; fundamental solution},
language = {eng},
number = {4},
pages = {1066-1087},
publisher = {EDP-Sciences},
title = {Estimate of the pressure when its gradient is the divergence of a measure. Applications},
url = {http://eudml.org/doc/272952},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Briane, Marc
AU - Casado-Díaz, Juan
TI - Estimate of the pressure when its gradient is the divergence of a measure. Applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 4
SP - 1066
EP - 1087
AB - In this paper, a $W^{-1,N^{\prime }}$ 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.
LA - eng
KW - pressure; Navier-Stokes equation; div-curl; measure data; fundamental solution
UR - http://eudml.org/doc/272952
ER -

References

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