Formation of singularities for multi-dimensional transport equations with nonlocal velocity

TL;DR

This paper investigates multi-dimensional transport equations with nonlocal velocity, demonstrating that smooth solutions cannot exist globally in time using the De Giorgi iteration method.

Contribution

It establishes the non-existence of global smooth solutions for a class of multi-dimensional nonlocal transport equations, advancing understanding of their singularity formation.

Findings

Local smooth solutions do not exist globally in time.

De Giorgi iteration technique is used to prove finite-time singularity formation.

Abstract

This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.