Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case

TL;DR

This paper proves the global existence, regularization, and long-term behavior of solutions for a non-local Burgers equation with periodic initial data, using advanced regularity theory for integro-differential equations.

Contribution

It establishes the global well-posedness and instant regularization of solutions for a non-local Burgers equation, extending the understanding of such equations in the periodic setting.

Findings

Global classical solutions from smooth positive data

Instantaneous regularization to smooth solutions

Description of long-time behavior of solutions

Abstract

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$ u_t - u |\nabla| u + |\nabla|(u^2) = 0. $$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.