Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

Nadia Belaribi, François Cuvelier and Francesco Russo
march, 2013
Publication type:
Paper in peer-reviewed journals
Stochastic Partial Differential Equations: Analysis and Computations, vol. 1 (1), pp. 3-62
Keywords :
Stochastic particle algorithm, porous media equation, monotonicity, stochastic differential equations, non-parametric density estimation, kernel estimator
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.
    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 },
    volume={1 (1) },