  This paper is concerned with the study of a nonlinear non-local equation that
has a commutator structure. The equation reads $\partial_t u-F(u)
(-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$
(0, 1]. We are interested in solutions stemming from periodic positive bounded
initial data. The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$
a.e. on (0, +$\infty$). For instance, all the functions $F (u) = u^n$ with n
$\in$ N * are admissible non-linearities. We construct global classical
solutions starting from smooth positive data, and global weak solutions
starting from positive data in $L^\infty$. We show that any weak solution is
instantaneously regularized into $C^\infty$. We also describe the long-time
asymptotics of all solutions. Our methods follow several recent advances in the
regularity theory of parabolic integro-differential equations.
