Journal

Annales de l’Institut Henri Poincaré C

Abstract

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential K. We are interested in establishing their well-posedness theory when the nonlocal interaction potential K is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that ∇K * K is in L², we can prove that the strong solution is unique. When K is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the n-species system.

Overview

One of the main motivations for this paper was the common use of top-hat kernels in nonlocal models. These kernels often appear in applications, but mathematically they are awkward because they are discontinuous. A big part of the paper is showing that you can still get a rigorous well-posedness theory in this rough setting by working with weaker regularity, such as bounded variation. In the scalar case, the paper splits into two main regimes. For merely bounded kernels, we work in a small-mass setting. If the kernel has additional structure (i.e., if it is even, so that an energy functional is available together with BV control), then we can also treat arbitrary mass. For systems, the picture is similar in spirit but a bit more delicate. There is again a small-mass regime, while the arbitrary-mass case relies on a detailed balance condition to recover the right energy structure. That condition is slightly weaker than a symmetry assumption (in terms of the nonlocal cross-interactions) that are often used in related models.

Keywords

aggregation-diffusion; irregular kernel; nonlocal PDE; entropy methods; cross-diffusion; multi-species models

MSC 2020

35K40 (primary); 35K55; 35A01; 35A02; 35B65; 35Q92

Links

• DOI / Journal page
• Preprint (arXiv)