no. 17-18
Partial Differential Equations/Optimal Control
A Hamilton–Jacobi PDE in the space of measures and its associated compressible Euler equations
[Une EDP de Hamilton–Jacobi dans lʼespace des mesures et ses équations dʼEuler compressibles associées]

Présenté par : the Editorial Board

Feng, Jin1
1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 973-976.

Nous introduisons une classe dʼintégrales dʼaction définies sur lʼespace des chemins à valeurs mesures de probabilité. Dans ce contexte lʼaction minimale existe et donne une solution faible dʼune équation dʼEuler compressible. Nous montrons que lʼéquation de Hamilton Jacobi associʼee à la formulation variationnelle de lʼéquation dʼEuler est bien posée dans le sens des solutions de viscosité.

We introduce a class of action integrals defined over probability-measure-valued path space. Minimal action exists in this context and gives weak solution to a compressible Euler equation. We prove that the Hamilton–Jacobi PDE associated with such variational formulation of Euler equation is well posed in viscosity solution sense.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.08.013
Feng, Jin 1

1 Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA 
@article{CRMATH_2011__349_17-18_973_0,
     author = {Feng, Jin},
     title = {A {Hamilton{\textendash}Jacobi} {PDE} in the space of measures and its associated compressible {Euler} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {973--976},
     publisher = {Elsevier},
     volume = {349},
     number = {17-18},
     year = {2011},
     doi = {10.1016/j.crma.2011.08.013},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2011.08.013/}
}
TY  - JOUR
AU  - Feng, Jin
TI  - A Hamilton–Jacobi PDE in the space of measures and its associated compressible Euler equations
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 973
EP  - 976
VL  - 349
IS  - 17-18
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2011.08.013/
DO  - 10.1016/j.crma.2011.08.013
LA  - en
ID  - CRMATH_2011__349_17-18_973_0
ER  - 
%0 Journal Article
%A Feng, Jin
%T A Hamilton–Jacobi PDE in the space of measures and its associated compressible Euler equations
%J Comptes Rendus. Mathématique
%D 2011
%P 973-976
%V 349
%N 17-18
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2011.08.013/
%R 10.1016/j.crma.2011.08.013
%G en
%F CRMATH_2011__349_17-18_973_0
Feng, Jin. A Hamilton–Jacobi PDE in the space of measures and its associated compressible Euler equations. Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 973-976. doi : 10.1016/j.crma.2011.08.013. http://archive.numdam.org/articles/10.1016/j.crma.2011.08.013/

[1] Ambrosio, L.; Gigli, G.N.; Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005

[2] Feng, J.; Katsoulakis, M. A comparison principle for Hamilton–Jacobi equations related to controlled gradient flows in infinite dimensions, Arch. Ration. Mech. Anal., Volume 192 (2009) no. 2, pp. 275-310

[3] Feng, J.; Kurtz, T. Large Deviation for Stochastic Processes, Mathematical Surveys and Monographs, vol. 131, American Mathematical Society, Providence, RI, 2006

[4] J. Feng, T. Nguyen, Hamilton–Jacobi equations in space of measures associated with a system of conservation laws, Preprint, 2010.

[5] Lasry, J.M.; Lions, P.-L. Mean field games, Japanese J. Math., Volume 2 (2007) no. 1, pp. 229-260

[6] Villani, C. Optimal Transport. Old and New, Fundamental Principles of Mathematical Sciences, vol. 338, Springer-Verlag, Berlin, 2009

Cité par Sources :