Global solutions to quadratic systems of stochastic reaction-diffusion equations in space-dimension two

Marta Leocata (1, 2) , Julien Vovelle (3)

1 SNS - Scuola Normale Superiore di Pisa
2 LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma]
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We prove the existence of global-in-time regular solutions to a system of stochastic quadratic reaction-diffusion equations. Global-in-time existence is based on a $L^\infty$-estimate obtained by an approach à la De Giorgi, as in [GoudonVasseur10]. The adaptation of this technique to the stochastic case requires in its final step an $L^2\ln(L^2)$-bound, furnished by an estimate by duality on the entropy inequality, as in [DesvillettesFellnerPierreVovelle07]. In our stochastic context, and similarly to [DebusscheRoselloVovelle2021], we need to solve a backward SPDE to exploit the duality technique

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Marta Leocata, Julien Vovelle. Global solutions to quadratic systems of stochastic reaction-diffusion equations in space-dimension two. 2023. ⟨hal-04283612v2⟩