Stochastic models associated to a Nonlocal Porous Medium Equation

Abstract

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights. Applications <February 5, 2019>is the fractional Laplace operator; i.e. a Fourier multiplier with the symbol ||ξ|| α . We observe that ∇ 1 = ∇ is the classical gradient and that div(Another equivalent definition of the fractional gradient ∇ α−1 involves the Riesz potential; that is ∇ α−1 = ∇I 2−α where I β = (−∆) − β 2 is a Fourier multiplier with symbol ||ξ|| −β , β ∈ (0, 2) (for more details on this point see [3,4]).In [3,4], explicit and compactly supported nonnegative self-similar solutions of (1) are constructed. These explicit solutions generalize the well-known Barenblatt-Kompanets-Zel'dovich-Pattle solutions of the porous medium equation (4) below. Furthermore, the authors proved the existence of sign-changing weak solution to the Cauchy problem (1)-(2) for u 0 (x) ∈ L 1 (R d ), and the hypercontractivity L 1 → L p estimates.By exploiting Darcy's law, it is possible to interpret the equation (1) as a transport equation ∂ t u = div(|u|v), where v := ∇p := ∇I 2−α (|u| m−2 u) is a vector velocity field with nonlocal and nonlinear pressure p in the case of nonnegative initial data. We observe that the fractional operator ∇I 2−α represents the long-range diffusion effects. The one-dimensional version of the pseudo-differential equation (1) describes the dynamics of dislocations in crystals (see [2]).For α = 2, (1) becomes the classical nonlinear porous medium equation

Random flights connecting porous medium and Euler–Poisson–Darboux equations

“…The same approach developed in this section has been applied to the probabilistic interpretation of a fractional version of the PME (see Ref. 27). Therefore, we start by introducing this class of stochastic processes.…”

Alternative probabilistic representations of Barenblatt-type solutions

“…Furthermore, u(x, t) is the pointwise solution of equation (2.4) for x = k − 1 ν t α . The link between (2.8) and random flights has been investigated in [8]. For ν = 2, the solution (2.3) becomes the Barenblatt-Kompanets-Zel'dovich-Pattle solution of the porous medium equation (2.7) supplemented with the initial condition u(x, 0) = δ(x) (see, for instance, [33]).…”