A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models

Hana Hajj Chehade (1) , Mustapha Jazar (2) , Régis Monneau (3)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
2 Mathematics Department
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We prove a priori gradient bounds for classical solutions of the fully nonlinear parabolic equation $$u_{t}=F(D^2u ,D u,u,x,t).$$ The domain is the torus {\mathbb{T}}^{d} of dimension $d\ge1$. Up to the price of technicalities, our work can be extended to the case of bounded domains or the case of the whole space ${\mathbb{R}}^d$. Several applications are given, including the standard porous medium equation.

Mots clés: maximum principle; viscosity solutions; nonlinear equations; degenerate parabolic equation; flows in porous medium; bounds for solutions; hydrology

Domaines: Equations aux dérivées partielles [math.AP]

Hana Hajj Chehade, Mustapha Jazar, Régis Monneau. A priori gradient bounds for fully nonlinear parabolic equations and applications to porous medium models. 2014. ⟨hal-00955311⟩