https://hal.inria.fr/hal-00723821Belaribi, NadiaNadiaBelaribiUMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques AvancéesLAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche ScientifiqueCuvelier, FrançoisFrançoisCuvelierLAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche ScientifiqueRusso, FrancescoFrancescoRussoOC - Optimisation et commande - UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques AvancéesENSTA Paris - École Nationale Supérieure de Techniques AvancéesProbabilistic and deterministic algorithms for space multidimensional irregular porous media equation.HAL CCSD2013Stochastic particle algorithmporous media equationmonotonicitystochastic differential equationsnon-parametric density estimationkernel estimator[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Russo, Francesco2012-08-20 10:44:262023-03-24 14:52:562012-08-20 11:06:05enJournal articleshttps://hal.inria.fr/hal-00723821/document10.1007/s40072-013-0001-7application/pdf1The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure.