On a transport equation with nonlocal drift

TL;DR

This paper investigates the nonlocal Burgers equation with Hilbert transform drift, providing multiple proofs of finite-time blow-up, exploring regularization effects, and analyzing related diffusive models with fractional Laplacian, extending understanding of nonlocal PDE behavior.

Contribution

It offers four different proofs of finite-time blow-up for the nonlocal Burgers equation and studies regularization effects in related diffusive models with fractional derivatives.

Findings

Nonlocal Burgers equation does not have global classical solutions for some initial data.

Regularization effects depend on the fractional diffusion parameter b3.

Solutions from vanishing viscosity limits are conjectured to be bounded in specific Hlder classes.

Abstract

In \cite{CordobaCordobaFontelos05}, C\'ordoba, C\'ordoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial_t \theta +u \; \partial_x \theta = 0, \qquad u = H \theta, \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible H\"older regularization effects of this equation and its consequences to the equation with diffusion \[ \partial_t \theta + u \; \partial_x \theta + \Lambda^\gamma \theta = 0, \qquad u = H \theta, \] where $\Lambda = (-\Delta)^{1/2}$, and $1/2 \leq \gamma <1$. Our results also apply to the model with velocity field $u = \Lambda^s H \theta$, where $s \in (-1,1)$. We conjecture that solutions which arise as limits from vanishing viscosity approximations are bounded in the H\"older class in $C^{(s+1)/2}$, for all positive time.