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)(T-3) give rise to a solution in C(R+; L-a,sigma(2)(T-3)) boolean AND L-2(R+; H-a,sigma(1)(T-3)), where L-a,sigma(2) is identified with the homogeneous Sobolev-Gevrey space. (H) over dot(a,sigma)(r) when r = 0 with parameters a is an element of (0, 1) and sigma >= 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.