Partial differential equations
The method of differential contractions
[La méthode des contractions différentielles]

Présenté par : the Editorial Board

Monneau, Régis1
1 CERMICS, École des ponts ParisTech, Université Paris-Est, 6 et 8, avenue Blaise-Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France
Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 143-147.

Dans cette Note, nous présentons une méthode simple et générale pour fabriquer des familles de contractions pour des équations aux dérivées partielles non linéaires, d'évolution, ou bien stationnaires. À titre d'exemple, cette méthode est appliquée à l'équation des milieux poreux, pour laquelle nous obtenons de nouvelles contractions. Cette méthode ouvre de nouvelles voies de recherche à explorer.

In this Note, we present a general and fairly simple method to design families of contractions for nonlinear partial differential equations, either of evolution type, or of stationary type. As a particular example, we apply this method to the porous medium equation, for which we get new contractions. This method opens new directions to explore.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.08.020
Monneau, Régis 1

1 CERMICS, École des ponts ParisTech, Université Paris-Est, 6 et 8, avenue Blaise-Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée cedex 2, France 
@article{CRMATH_2015__353_2_143_0,
     author = {Monneau, R\'egis},
     title = {The method of differential contractions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {143--147},
     publisher = {Elsevier},
     volume = {353},
     number = {2},
     year = {2015},
     doi = {10.1016/j.crma.2014.08.020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.08.020/}
}
TY  - JOUR
AU  - Monneau, Régis
TI  - The method of differential contractions
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 143
EP  - 147
VL  - 353
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2014.08.020/
DO  - 10.1016/j.crma.2014.08.020
LA  - en
ID  - CRMATH_2015__353_2_143_0
ER  - 
%0 Journal Article
%A Monneau, Régis
%T The method of differential contractions
%J Comptes Rendus. Mathématique
%D 2015
%P 143-147
%V 353
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2014.08.020/
%R 10.1016/j.crma.2014.08.020
%G en
%F CRMATH_2015__353_2_143_0
Monneau, Régis. The method of differential contractions. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 143-147. doi : 10.1016/j.crma.2014.08.020. http://www.numdam.org/articles/10.1016/j.crma.2014.08.020/

[1] Benilan, P.; Brezis, H.; Crandall, M.G. A semilinear equation in L1(RN), Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 2 (1975) no. 4, pp. 523-555

[2] Carrillo, J.A.; Jüngel, A.; Markowich, P.A.; Toscani, G.; Unterreiter, A. Entropy dissipation methods for degenerate parabolic problems and generalized sobolev inequalities, Monatshefte Math., Volume 133 (2001), pp. 1-82

[3] Chmaycem, G. Study of the porous medium equation and of a blister model, École des ponts ParisTech, Paris, 2014 (PhD thesis)

[4] G. Chmaycem, M. Jazar, R. Monneau, A new contraction family for porous medium and fast diffusion equations, in preparation.

[5] G. Chmaycem, M. Jazar, R. Monneau, in preparation.

[6] Del Pino, M.; Dolbeault, J. Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), Volume 81 (2002), pp. 847-875

[7] Desvillettes, L.; Villani, C. Entropic methods for the study of the long time behavior of kinetic equations, Transp. Theory Stat. Phys., Volume 30 (2001) no. 2, 3, pp. 155-168

[8] Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Commun. Partial Differ. Equ., Volume 26 (2001) no. 1–2, pp. 101-174

[9] Sacks, P.E. Continuity of solutions of a singular parabolic equation, Nonlinear Anal., Volume 7 (1983), pp. 387-409

[10] Vázquez, J.-L. The porous medium equation: new contractivity results, Progr. Nonlinear Differential Equations Appl., Volume 63 (2005), pp. 433-451

[11] Vázquez, J.-L. Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and Its Applications, 2006

[12] Vázquez, J.-L. The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 2007

Cité par Sources :