Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads

$$ \partial _{t} u-F(u) \, (-\Delta )^{s/{2}} u+(-\Delta )^{s/{2}} (uF(u))=0, \quad x\in \mathbb{T}^{d}, $$

with \(s\in (0, 1]\). We are interested in solutions stemming from periodic positive bounded initial data. The given function \(F\in \mathcal{C}^{\infty }(\mathbb{R}^{+})\) must satisfy \(F'>0\) a.e. on \((0, +\infty )\). For instance, all the functions \(F(u)=u^{n}\) with \(n\in \mathbb{N}^{\ast }\) are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. 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 \(\mathcal{C}^{\infty }\). 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, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018).