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2016 Local well-posedness of a nonlocal Burgers' equation
Sam Goodchild, Hang Yang
Involve 9(1): 67-82 (2016). 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 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.

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Sam Goodchild. Hang Yang. "Local well-posedness of a nonlocal Burgers' equation." Involve 9 (1) 67 - 82, 2016. https://doi.org/10.2140/involve.2016.9.67

Information

Received: 16 September 2013; Revised: 6 June 2014; Accepted: 8 June 2014; Published: 2016
First available in Project Euclid: 22 November 2017

zbMATH: 1332.35077
MathSciNet: MR3438446
Digital Object Identifier: 10.2140/involve.2016.9.67

Subjects:
Primary: 35F20

Keywords: finite-time blow-up , global regularity , nonlocal Burgers' equation

Rights: Copyright © 2016 Mathematical Sciences Publishers

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