Formation of singularities for a transport equation with nonlocal velocity

TL;DR

This paper investigates how singularities form in a one-dimensional transport equation with nonlocal velocity and demonstrates that adding diffusion prevents singularity formation, ensuring global existence of solutions.

Contribution

It introduces a detailed analysis of singularity formation in a nonlocal transport equation and shows how diffusion influences solution behavior.

Findings

Singularities form in finite time for certain initial data.

Adding diffusion prevents finite time singularities.

Solutions exist globally when diffusion is present.

Abstract

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.