On the local and global existence of solutions to 1D transport equations with nonlocal velocity

TL;DR

This paper investigates the existence of solutions to a one-dimensional transport equation with a nonlocal velocity field, establishing conditions for both local and global solutions depending on the type of nonlocal operator involved.

Contribution

The paper provides new results on the existence of solutions to 1D transport equations with various nonlocal velocity operators, extending previous work to broader classes of nonlocal interactions.

Findings

Existence of local solutions for certain nonlocal operators.

Global solutions established under specific conditions.

Applicable to a range of nonlocal velocity operators.

Abstract

We consider the 1D transport equation with nonlocal velocity field: \begin{equation*}\label{intro eq} \begin{split} &\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta=0, \\ & u=\mathcal{N}(\theta), \end{split} \end{equation*} where $\mathcal{N}$ is a nonlocal operator. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.