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

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$u_t-u\mathrm{\mid }\mathrm{\nabla }\mathrm{\mid }u+\mathrm{\mid }\mathrm{\nabla }\mathrm{\mid }(u^2)=0.$$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^{\mathrm{\infty }}$. We show that any weak solution is instantaneously regularized into $C^{\mathrm{\infty }}$. 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.