Well-posedness, stability and determining modes to 3D Burgers equation in Gevrey class

Abstract

This paper aims to prove that the three-dimensional periodic Burgers equation has a unique global in time solution, in the Lebesgue–Gevrey space. In particular, the initial data that belong to $L_{a,\sigma }^{2}(\mathbb {T}^3)$ give rise to a solution in $C(\mathbb {R}_+;L_{a,\sigma }^{2}(\mathbb {T}^3))\cap L^2(\mathbb {R}_+;H_{a,\sigma }^{1}(\mathbb {T}^3))$, where $L_{a,\sigma }^2$ is identified with the homogeneous Sobolev–Gevrey space $\dot{H}_{a,\sigma }^r$ when $r=0$ with parameters $a \in (0,1)$ and $\sigma \ge 1$. We also prove that the solution is stable under perturbation and that the long-time behavior of Burgers system is determined by a finite number of degrees of freedom in $L_{a,\sigma }^2$. Energy methods, compactness methods and Fourier analysis are the main tools.