Estimate of the pressure when its gradient is the divergence of a measure. Applications
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 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.

DOI : https://doi.org/10.1051/cocv/2010037
Classification:  35Q30,  35Q35,  35A08
Keywords: pressure, Navier-Stokes equation, div-curl, measure data, fundamental solution
@article{COCV_2011__17_4_1066_0,
author = {Briane, Marc and Casado-D\'\i az, Juan},
title = {Estimate of the pressure when its gradient is the divergence of a measure. Applications},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
pages = {1066-1087},
doi = {10.1051/cocv/2010037},
zbl = {1232.35113},
mrnumber = {2859865},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_4_1066_0}
}

Briane, Marc; Casado-Díaz, Juan. Estimate of the pressure when its gradient is the divergence of a measure. Applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1066-1087. doi : 10.1051/cocv/2010037. http://www.numdam.org/item/COCV_2011__17_4_1066_0/

[1] 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. | MR 1257940 | Zbl 0823.35140

[2] 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. | Numdam | MR 1635769 | Zbl 0919.35014

[3] 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. | MR 553920 | Zbl 0499.35022

[4] J. Bourgain and H. Brezis, New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc. 9 (2007) 277-315. | MR 2293957 | Zbl 1176.35061

[5] H. Brezis, Analyse Fonctionnelle, Théorie et Applications. Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983). | MR 697382 | Zbl 0511.46001

[6] H. Brezis and J. Van Schaftingen, Boundary estimates for elliptic systems with L1-data. Calc. Var. 30 (2007) 369-388. | MR 2332419 | Zbl 1149.35025

[7] M. Briane, Homogenization of the Stokes equations with high-contrast viscosity. J. Math. Pures Appl. 82 (2003) 843-876. | MR 2005297 | Zbl 1058.35024

[8] 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. | MR 2438743 | Zbl 1167.35336

[9] M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals. Arch. Rat. Mech. Anal. 170 (2003) 211-245. | MR 2020260 | Zbl 1030.74013

[10] G. De Rham, Variétés différentiables, Formes, courants, formes harmoniques. Hermann, Paris (1973). | Zbl 0284.58001

[11] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001). | MR 1814364 | Zbl 1042.35002

[12] E. Hopf, Über die Anfwangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951) 213-231. | MR 50423 | Zbl 0042.10604

[13] 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. | MR 1145750 | Zbl 0737.73009

[14] 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). | MR 254401 | Zbl 0121.42701

[15] J.-L. Lions, Quelques résultats d'existence dans des équations aux dérivées partielles non linéaires. Bull. S.M.F. 87 (1959) 245-273. | Numdam | MR 115019 | Zbl 0147.07902

[16] V.A. Marchenko and E.Y. Khruslov, Homogenization of partial differential equations, Progress in Mathematical Physics 46. Birkhäuser, Boston (2006). | MR 2182441 | Zbl 1113.35003

[17] J. Nečas, Équations aux dérivées partielles. Presses de l'Université de Montréal (1965). | Zbl 0147.07801

[18] 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. | MR 1482641 | Zbl 0893.73006

[19] 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. | MR 341064 | Zbl 0259.35008

[20] L. Tartar, Topics in nonlinear analysis. Publications Mathématiques d'Orsay 78 (1978) 271. | MR 532371 | Zbl 0395.00008

[21] R. Temam, Navier-Stokes Equations - Theory and Numerical Analysis, Studies in Mathematics and its Applications 2. North-Holland, Amsterdam (1984). | Zbl 0568.35002

[22] J. Van Schaftingen, Estimates for L1-vector fields under higher-order differential conditions. J. Eur. Math. Soc. 10 (2008) 867-882. | MR 2443922 | Zbl 1228.46034

[23] J. Van Schaftingen, Estimates for L1-vector fields. C. R. Acad. Sci. Paris, Ser. I 339 (2004) 181-186. | MR 2078071 | Zbl 1049.35069