On a 1D nonlocal transport equation with nonlocal velocity and subcritical or supercritical diffusion

TL;DR

This paper investigates a 1D nonlocal transport equation with nonlocal velocity and fractional dissipation, establishing global existence in the subcritical case and local existence for large data in the supercritical case using advanced harmonic analysis techniques.

Contribution

It provides the first global existence results for subcritical dissipation and local existence for large data in the supercritical regime for this class of nonlocal transport equations.

Findings

Global existence in subcritical case $eta o 2$

Local existence for large data in supercritical case $eta o 0$

Use of weighted Littlewood-Paley theory and commutator estimates

Abstract

We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $H^{k}(w_{\lambda,\kappa}) \cap L^{\infty},$ with $k=\max(0,3/2-\alpha)$ and $w_{\lambda, \kappa}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $\alpha \in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{\lambda, \kappa})\cap L^{\infty}$ in the supercritical case $\alpha \in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.