Global solutions of aggregation equations and other flows with random diffusion

TL;DR

This paper investigates whether adding random diffusion to aggregation and active scalar equations can ensure global solutions, extending stochastic regularization results to a broad class of models including Hamiltonian and gradient flows.

Contribution

It demonstrates that suitable random diffusion can restore global existence for various active scalar equations with possibly singular velocity fields in any dimension.

Findings

Global solutions exist with high probability under random diffusion.

Applicable to Hamiltonian and gradient flow models, including SQG.

Solutions are shown in Gevrey-type Fourier-Lebesgue spaces.

Abstract

Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. arXiv:1806.03734 showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability.