A Discrete Model for Nonlocal Transport Equations with Fractional Dissipation

TL;DR

This paper introduces a discrete model for one-dimensional nonlocal transport equations with fractional dissipation, demonstrating blow-up behavior and regularity results that inform the understanding of the continuous equations.

Contribution

The paper develops a novel discrete model for nonlocal transport equations with fractional dissipation and establishes blow-up and regularity results analogous to the continuous case.

Findings

Blow-up results similar to continuous equations.

Regularity in the supercritical range $1/4<eta<1/2$.

Insights into supercritical dissipation effects.

Abstract

In this note, we propose a discrete model to study one-dimensional transport equations with non-local drift and supercritical dissipation. The inspiration for our model is the equation $$ \theta_t + (H\theta) \theta_x +(-\Delta)^\alpha \theta =0 $$ where $H$ is the Hilbert transform. For our discrete model, we present blow-up results that are analogous to the known results for the above equation. In addition, we will prove regularity for our discrete model which suggests supercritical regularity in the range $1/4<\alpha<1/2$ in the continuous setting.