Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation. - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Stochastics and Partial Differential Equations: Analysis and Computations Year : 2013

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

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.
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Dates and versions

hal-00723821 , version 1 (20-08-2012)

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Cite

Nadia Belaribi, François Cuvelier, Francesco Russo. Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation.. Stochastics and Partial Differential Equations: Analysis and Computations, 2013, 1 (1), pp.3-62. ⟨10.1007/s40072-013-0001-7⟩. ⟨hal-00723821⟩
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