We study the asymptotics of Allen–Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i) the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii) the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.
Révisé le :
Accepté le :
Publié le :
Pierre-Louis Lions 1 ; Panagiotis E. Souganidis 2
CC-BY 4.0
@article{CRMATH_2020__358_8_931_0,
author = {Pierre-Louis Lions and Panagiotis E. Souganidis},
title = {The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation},
journal = {Comptes Rendus. Math\'ematique},
pages = {931--938},
publisher = {Acad\'emie des sciences, Paris},
volume = {358},
number = {8},
year = {2020},
doi = {10.5802/crmath.117},
language = {en},
}
TY - JOUR AU - Pierre-Louis Lions AU - Panagiotis E. Souganidis TI - The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation JO - Comptes Rendus. Mathématique PY - 2020 SP - 931 EP - 938 VL - 358 IS - 8 PB - Académie des sciences, Paris DO - 10.5802/crmath.117 LA - en ID - CRMATH_2020__358_8_931_0 ER -
%0 Journal Article %A Pierre-Louis Lions %A Panagiotis E. Souganidis %T The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation %J Comptes Rendus. Mathématique %D 2020 %P 931-938 %V 358 %N 8 %I Académie des sciences, Paris %R 10.5802/crmath.117 %G en %F CRMATH_2020__358_8_931_0
Pierre-Louis Lions; Panagiotis E. Souganidis. The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 931-938. doi : 10.5802/crmath.117. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.117/
[1] Generation of fine transition layers and their dynamics for the stochastic Allen–Cahn equation (2018) (https://arxiv.org/abs/1812.03804)
[2] Front propagation and phase field theory, SIAM J. Control Optimization, Volume 31 (1993) no. 2, pp. 439-469 | DOI | MR | Zbl
[3] A new approach to front propagation problems: theory and applications, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 3, pp. 237-296 | DOI | MR | Zbl
[4] Generation and propagation of interfaces for reaction-diffusion equations, J. Differ. Equations, Volume 96 (1992) no. 1, pp. 116-141 | DOI | MR | Zbl
[5] Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., Volume 45 (1992) no. 9, pp. 1097-1123 | DOI | MR | Zbl
[6] Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin., Engl. Ser., Volume 15 (1999) no. 3, pp. 407-438 | DOI | MR | Zbl
[7] Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland, 1992 | Zbl
[8] (in preparation) | DOI | Numdam
[9] Fully nonlinear stochastic partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 326 (1998) no. 9, pp. 1085-1092 | DOI | MR | Zbl
[10] Fully nonlinear stochastic partial differential equations: nonsmooth equations and applications, C. R. Math. Acad. Sci. Paris, Volume 327 (1998) no. 8, pp. 735-741 | DOI | Zbl
[11] Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Math. Acad. Sci. Paris, Volume 331 (2000) no. 10, pp. 783-790 | DOI | MR | Zbl
[12] Universal scaling in the motion of random interfaces, Phys. Rev. Lett., Volume 49 (1982) no. 17, pp. 1223-1226 | DOI
[13] Pathwise solutions for fully nonlinear first- and second-order partial differential equations with multiplicative rough time dependence, Singular Random Dynamics (Lecture Notes in Mathematics), Volume 2253, Springer, 2019 | DOI | MR
[14] Stochastic motion of mean curvature, Arch. Ration. Mech. Anal., Volume 144 (1998) no. 4, pp. 313-355 | MR | Zbl
Cité par Sources :
Commentaires - Politique