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 ut +(u(x + h,t) ± u(x h,t))ux = 0 with given periodic initial condition u(x,0) = u0(x). 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
Mathematical Subject Classification 2010
Primary: 35F20
Milestones
Received: 16 September 2013
Revised: 6 June 2014
Accepted: 8 June 2014
Published: 17 December 2015

Communicated by Martin Bohner
Authors
Sam Goodchild
University of Wisconsin–Madison
Madison, WI 53706
United States
Hang Yang
Department of Mathematics
Rice University
Houston, TX 77005
United States