#### Vol. 9, No. 1, 2016

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Local well-posedness of a nonlocal Burgers' equation

### Sam Goodchild and Hang Yang

Vol. 9 (2016), No. 1, 67–82
##### 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 ${u}_{t}+\left(u\left(x+h,t\right)±u\left(x-h,t\right)\right){u}_{x}=0$ 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.

##### Keywords
nonlocal Burgers' equation, finite-time blow-up, global regularity
Primary: 35F20