On a 1D transport equation with nonlocal velocity and supercritical dissipation

TL;DR

This paper investigates a one-dimensional transport equation with nonlocal velocity, demonstrating eventual regularization and global regularity of solutions under supercritical dissipation using a nonlocal maximum principle.

Contribution

It establishes new regularity results for a 1D nonlocal transport equation with supercritical dissipation, extending understanding of solution behavior.

Findings

Eventual regularization of viscous solutions with non-negative initial data.

Global regularity for solutions with slightly supercritical dissipation.

Use of a nonlocal maximum principle to prove regularity results.

Abstract

We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle.