Porous medium equation as limit of nonlocal interaction

Abstract

This paper studies the convergence of solutions of a nonlocal interaction equation to the solution of the quadratic porous medium equation in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use the gradient flow structure of the equations to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme which could be slightly modified in order to extend our result to PDEs with no gradient flow structure. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy. However, the latter excludes particle solutions of the nonlocal interaction equation.

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

“…The same result is proven also for (NLE) and (DE). In particular, this extends [7] to the case m > 2, which is not trivial in view of the nonlinearities involved, and to a class of general nonlinear diffusion function. In [10], their gradient flow convergence result for m > 2 was conditional on a uniform BV bound for ρ ε while we make no such assumptions here.…”

“…Recent works in the literature show a rigorous and fascinating connection between the two energies above for m > 1 in (1.2) and the corresponding dynamics, by means of gradient flow techniques, c.f. [10,7]. More precisely, exploiting the so-called blob method developed in [25], one can notice already at a formal level that an appropriate regularisation of H m transforms a diffusion equation (which is local) into an interaction PDE (which is nonlocal) by choosing a delocalising kernel.…”

“…which motivates the consideration of (NLE) as the 2-Wasserstein gradient flow of F ε . Following the strategy proposed in [7], defining the pressure by P (x) := xF ′ (x) − F (x), as in [41,19,1], we construct weak solutions of the nonlinear diffusion equation…”

“…In the case m = 2, in the same spirit of [10,26], the authors in [7] construct weak solutions of the quadratic porous medium equation as a localising limit (ε → 0) of a sequence of weak measure solutions of the nonlocal interaction equation (NLE), for F (x) = x 2 . The authors work directly at the level of the (nonlocal) equations by means of a time-discretisation scheme which allows to work with lack of convexity, as for instance in the case of cross-diffusion systems, or even PDEs with no purely gradient flow structure.…”

“…As in [10], finite initial log-entropy is required, thus excluding particle approximation. However, simultaneously to [7], the authors in [26] focus on a weighted (quadratic) porous medium equation which is relevant, e.g. in sampling -the weight, ρ in their notations, represents a target probability measure to be approximated from specific samples drawn from it.…”