On a multi-dimensional transport equation with nonlocal velocity and fractional dissipation

TL;DR

This paper investigates a multi-dimensional transport equation with nonlocal velocity and fractional dissipation, analyzing subcritical, critical, and supercritical cases to establish well-posedness, regularity, and blowup results.

Contribution

It provides a comprehensive analysis of the equation's behavior across different regimes, including new results on well-posedness and blowup.

Findings

Local well-posedness established

Global smoothness in certain regimes

Finite-time blowup in others

Abstract

This paper aims to investigate a multi-dimensional transport equation with nonlocal velocity and fractional dissipation.The balance between the nonlinearity and dissipation gives rise to three different cases, namely the subcritical, critical and supercritical ranges. We study those three cases and obtain a set of results containing local well-posedness, global smoothness, eventual regularity and finite-time blowup of smooth solutions.