Local well-posedness of a nonlocal Burgers' equation

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 $u_t+\left(u\left(x+h,t\right)\pm u\left(x-h,t\right)\right)u_x=0$ with given periodic initial condition $u\left(x,0\right)=u_0\left(x\right)$. We also explore the blow-up properties of the solutions to this Cauchy problem, and show that there exist initial data that lead to finite-time-blow-up solutions and others to globally regular solutions. This contrasts with the classical inviscid Burgers’ equation, for which all nonconstant smooth periodic initial data lead to finite-time blow-up. Finally, we present results of simulations to illustrate our findings.