Global well-posedness and long-time asymptotics of a general nonlinear non-local Burgers Equation

Abstract

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 $F^{\prime }>0$ 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.