# Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

march, 2013

Publication type:

Paper in peer-reviewed journals

Journal:

Stochastic Partial Differential Equations: Analysis and Computations, vol. 1 (1), pp. 3-62

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Keywords :

Stochastic particle algorithm, porous media equation,
monotonicity, stochastic differential equations,
non-parametric density estimation, kernel estimator

Abstract:

The 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.

BibTeX:

@article{Bel-Cuv-Rus-2013, author={Nadia Belaribi and François Cuvelier and Francesco Russo }, title={Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation }, doi={10.1007/s40072-013-0001-7 }, journal={Stochastic Partial Differential Equations: Analysis and Computations }, year={2013 }, month={3}, volume={1 (1) }, pages={3--62}, }