A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time

Guy Barles

doi:10.1016/j.crma.2006.06.022

Partial Differential Equations

A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time
[Un nouveau résultat de stabilité pour les solutions de viscosité d'équations paraboliques non-linéaires avec convergence faible en temps]

Présenté par : Pierre-Louis Lions

Guy Barles ¹

¹ Laboratoire de mathématiques et physique théorique (UMR CNRS 6083), fédération Denis-Poisson, université de Tours, parc de Grandmont, 37200 Tours, France

Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 173-178.

Résumé
Abstract

Nous obtenons un nouveau résultat de stabilité pour les solutions de viscosité d'équations fortement non linéaires paraboliques dans le cas où l'on n'a qu'une convergence faible en temps pour les non-linéarités.

We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities.

Reçu le : 2006-01-08
Accepté le : 2006-06-01
Publié le : 2006-07-11

DOI : 10.1016/j.crma.2006.06.022

Affiliations des auteurs :

Guy Barles ¹

¹ Laboratoire de mathématiques et physique théorique (UMR CNRS 6083), fédération Denis-Poisson, université de Tours, parc de Grandmont, 37200 Tours, France

@article{CRMATH_2006__343_3_173_0,
     author = {Guy Barles},
     title = {A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {173--178},
     publisher = {Elsevier},
     volume = {343},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2006.06.022},
     language = {en},
}

TY  - JOUR
AU  - Guy Barles
TI  - A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 173
EP  - 178
VL  - 343
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2006.06.022
LA  - en
ID  - CRMATH_2006__343_3_173_0
ER  -

%0 Journal Article
%A Guy Barles
%T A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time
%J Comptes Rendus. Mathématique
%D 2006
%P 173-178
%V 343
%N 3
%I Elsevier
%R 10.1016/j.crma.2006.06.022
%G en
%F CRMATH_2006__343_3_173_0

Guy Barles. A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time. Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 173-178. doi : 10.1016/j.crma.2006.06.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.022/

Bibliographie
Cité par

[1] M. Bardi; I. Capuzzo-Dolcetta Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997

[2] G. Barles; C. Georgelin A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., Volume 32 (1995) no. 2, pp. 484-500

[3] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions, Preprint

[4] M. Bourgoing, Viscosity solutions of fully nonlinear second order parabolic equations with L1-time dependence and Neumann boundary conditions, Existence and applications to the level-set approach, Preprint

[5] M.G. Crandall; H. Ishii; P.L. Lions User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), Volume 27 (1992) no. 1, pp. 1-67

[6] W.H. Fleming; H.M. Soner Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics (New York), vol. 25, Springer-Verlag, New York, 1993

[7] H. Ishii Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets, Bull. Fac. Sci. Engrg. Chuo Univ., Volume 28 (1985), pp. 33-77

[8] P.L. Lions; B. Perthame Remarks on Hamilton–Jacobi equations with measurable time-dependent Hamiltonians, Non-linear Analysis. Theory Methods and Applications, Volume 11 (1987), pp. 613-621

[9] P.L. Lions; P.E. Souganidis Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998) no. 9, pp. 1085-1092

[10] P.L. Lions; P. E Souganidis Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math., Volume 327 (1998) no. 8, pp. 735-741

[11] P.L. Lions, P.E. Souganidis, personal communications, various courses and lectures

[12] D. Nunziante Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence, Differential and Integral Equations, Volume 3 (1990) no. 1, pp. 77-91

[13] D. Nunziante Existence and uniqueness of viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Analysis. Theory, Methods and Applications, Volume 18 (1992) no. 11, pp. 1033-1062

Teo Kukuljan Higher order parabolic boundary Harnack inequality in $C^{1}$ and $C^{k, α}$ domains, Discrete and Continuous Dynamical Systems, Volume 42 (2022) no. 6, pp. 2667-2698 | DOI:10.3934/dcds.2021207 | Zbl:1490.35061
Shota Tateyama Hölder gradient estimates on $L^{p}$ -viscosity solutions of fully nonlinear parabolic equations with VMO coefficients, SN Partial Differential Equations and Applications, Volume 2 (2021) no. 6, p. 22 (Id/No 81) | DOI:10.1007/s42985-021-00133-4 | Zbl:1491.35085
Antonin Chambolle; Massimiliano Morini; Matteo Novaga; Marcello Ponsiglione Generalized crystalline evolutions as limits of flows with smooth anisotropies, Analysis PDE, Volume 12 (2019) no. 3, pp. 789-813 | DOI:10.2140/apde.2019.12.789 | Zbl:1403.53055
Pierre Cardaliaguet; Charles-Albert Lehalle Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics, Volume 12 (2018) no. 3, pp. 335-363 | DOI:10.1007/s11579-017-0206-z | Zbl:1397.91084
A. El Hajj Global solution for a non-local eikonal equation modelling dislocation dynamics, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 168 (2018), pp. 154-175 | DOI:10.1016/j.na.2017.11.012 | Zbl:1383.35050
R. Boudjerada; A. El Hajj Global existence results for eikonal equation with $B V$ initial data, NoDEA. Nonlinear Differential Equations and Applications, Volume 22 (2015) no. 4, pp. 947-978 | DOI:10.1007/s00030-015-0310-9 | Zbl:1327.35066
Paul Gassiat; Harald Oberhauser; Gonçalo dos Reis Root's barrier, viscosity solutions of obstacle problems and reflected FBSDEs, Stochastic Processes and their Applications, Volume 125 (2015) no. 12, pp. 4601-4631 | DOI:10.1016/j.spa.2015.07.010 | Zbl:1335.60059
Zdzisław Brzeźniak; Ben Goldys; Misha Neklyudov Multidimensional Stochastic Burgers Equation, SIAM Journal on Mathematical Analysis, Volume 46 (2014) no. 1, p. 871 | DOI:10.1137/120866117
Guy Barles; Olivier Ley; Hiroyoshi Mitake Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations, Mathematische Annalen, Volume 352 (2012) no. 2, pp. 409-451 | DOI:10.1007/s00208-011-0648-1 | Zbl:1246.35013
Ariela Briani; Hasnaa Zidani Characterization of the value function of final state constrained control problems with BV trajectories, Communications on Pure and Applied Analysis, Volume 10 (2011) no. 6, p. 1567 | DOI:10.3934/cpaa.2011.10.1567
N. Forcadel; Z. Rao; H. Zidani Optimal control problems of BV trajectories with pointwise state constraints, IFAC Proceedings Volumes, Volume 44 (2011) no. 1, p. 2583 | DOI:10.3182/20110828-6-it-1002.01694
V. S. Borkar; K. Suresh Kumar A new Markov selection procedure for degenerate diffusions, Journal of Theoretical Probability, Volume 23 (2010) no. 3, pp. 729-747 | DOI:10.1007/s10959-009-0242-6 | Zbl:1214.60022
Aurélien Monteillet Convergence of approximation schemes for nonlocal front propagation equations, Mathematics of Computation, Volume 79 (2010) no. 269, pp. 125-146 | DOI:10.1090/s0025-5718-09-02270-4 | Zbl:1208.65138
Ben Goldys; Misha Neklyudov Beale-Kato-Majda type condition for Burgers equation, Journal of Mathematical Analysis and Applications, Volume 354 (2009) no. 2, pp. 397-411 | DOI:10.1016/j.jmaa.2008.12.043 | Zbl:1173.35069
Guy Barles; Pierre Cardaliaguet; Olivier Ley; Aurélien Monteillet Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 71 (2009) no. 7-8, pp. 2801-2810 | DOI:10.1016/j.na.2009.01.156 | Zbl:1166.49027
Guy Barles; Pierre Cardaliaguet; Olivier Ley; Régis Monneau Global Existence Results and Uniqueness for Dislocation Equations, SIAM Journal on Mathematical Analysis, Volume 40 (2008) no. 1, p. 44 | DOI:10.1137/070682083

Cité par 16 documents. Sources : Crossref, zbMATH

Commentaires - Politique