Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient β, which is well-posed as an evolution problem in L¹ (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 β 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 β is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that β 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.