Local Well-posedness of A Non-local Burgers Equation

TL;DR

This paper investigates a nonlocal inviscid Burgers equation, establishing local well-posedness, analyzing blow-up behavior, and contrasting it with classical Burgers dynamics, supported by numerical simulations.

Contribution

It proves local existence and uniqueness for the nonlocal Burgers equation and explores blow-up phenomena, highlighting differences from the classical case.

Findings

Existence and uniqueness of local solutions

Presence of finite-time blow-up solutions

Existence of globally regular solutions

Abstract

In this paper, we explore a nonlocal inviscid Burgers equation. Fixing a parameter $h$, we prove existence and uniqueness of the local solution of the equation $\InviscidBurgersNonlocal{u}$ with periodic initial condition. We also explore the blow up properties of solutions to these kinds of equations with given periodic initial data, and show that there exists solutions that blow up in finite time and solutions that are globally regular. This contrasts with the classical inviscid Burgers equation, for which all non-constant smooth periodic initial data lead to finite time blow up. Finally, we present results of simulations to illustrate our findings.