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

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

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

## Access Full Article

top## Abstract

top## How to cite

topBriane, 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/221898>.

@article{Briane2011,

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.
},

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; pressure},

language = {eng},

month = {11},

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/221898},

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

DA - 2011/11//

PB - EDP Sciences

VL - 17

IS - 4

SP - 1066

EP - 1087

AB -
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.

LA - eng

KW - Pressure; Navier-Stokes equation; div-curl; measure data; fundamental solution; pressure

UR - http://eudml.org/doc/221898

ER -

## References

top- C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J.44 (1994) 109–140. Zbl0823.35140
- M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects. Ann. Scuola Norm. Sup. Pisa Cl. Sci.26 (1998) 407–436. Zbl0919.35014
- M.E. Bogovski, Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Soviet Math. Dokl.20 (1979) 1094–1098. Zbl0499.35022
- J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc.9 (2007) 277–315. Zbl1176.35061
- H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983).
- H. Brezis and J. Van Schaftingen, Boundary estimates for elliptic systems with L1-data. Calc. Var.30 (2007) 369–388. Zbl1149.35025
- M. Briane, Homogenization of the Stokes equations with high-contrast viscosity. J. Math. Pures Appl.82 (2003) 843–876. Zbl1058.35024
- M. Briane and J. Casado Díaz, Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects. Calc. Var.33 (2008) 463–492. Zbl1167.35336
- M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals. Arch. Rat. Mech. Anal.170 (2003) 211–245. Zbl1030.74013
- G. de Rham, Variétés différentiables, Formes, courants, formes harmoniques. Hermann, Paris (1973). Zbl0284.58001
- D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). Zbl1042.35002
- E. Hopf, Über die Anfwangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr.4 (1951) 213–231. Zbl0042.10604
- E.Y. Khruslov, Homogenized models of composite media, in Composite Media and Homogenization Theory, G. Dal Maso and G.F. Dell'Antonio Eds., Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser (1991) 159–182. Zbl0737.73009
- O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications 2. Gordon and Breach, Science Publishers, New York-London-Paris (1969).
- J.-L. Lions, Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires. Bull. S.M.F.87 (1959) 245–273. Zbl0147.07902
- V.A. Marchenko and E.Y. Khruslov, Homogenization of partial differential equations, Progress in Mathematical Physics46. Birkhäuser, Boston (2006). Zbl1113.35003
- J. Nečas, Équations aux dérivées partielles. Presses de l'Université de Montréal (1965).
- C. Pideri and P. Seppecher, A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech. Thermodyn.9 (1997) 241–257. Zbl0893.73006
- M.-J. Strauss, Variations of Korn's and Sobolev's inequalities, in Partial Differential Equations: Proc. Symp. Pure Math.23, D. Spencer Ed., Am. Math. Soc., Providence (1973) 207–214. Zbl0259.35008
- L. Tartar, Topics in nonlinear analysis. Publications Mathématiques d'Orsay78 (1978) 271. Zbl0395.00008
- R. Temam, Navier-Stokes Equations – Theory and Numerical Analysis, Studies in Mathematics and its Applications2. North-Holland, Amsterdam (1984). Zbl0568.35002
- J. Van Schaftingen, Estimates for L1-vector fields under higher-order differential conditions. J. Eur. Math. Soc.10 (2008) 867–882. Zbl1228.46034
- J. Van Schaftingen, Estimates for L1-vector fields. C. R. Acad. Sci. Paris, Ser. I339 (2004) 181–186. Zbl1049.35069

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.