  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.
