Local well-posedness of a nonlocal Burgers’ equation

Goodchild, Sam; Yang, Hang

doi:10.2140/involve.2016.9.67

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Local well-posedness of a nonlocal Burgers' equation

Sam Goodchild and Hang Yang

Vol. 9 (2016), No. 1, 67–82

DOI: 10.2140/involve.2016.9.67

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 $u_{t} + (u (x + h, t) \pm u (x - h, t)) u_{x} = 0$ with given periodic initial condition $u (x, 0) = u_{0} (x)$ . 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

Vol. 9, No. 1, 2016