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.

本文旨在证明在Lebesgue-Gevrey空间中，三维周期Burgers方程具有唯一的全局时间解。特别是，属于\（L_ {a，\ sigma} ^ {2}（\ mathbb {T} ^ 3）\）的初始数据在\（C（\ mathbb {R} _ + ; L_ {a，\ sigma} ^ {2}（\ mathbb {T} ^ 3））\ cap L ^ 2（\ mathbb {R} _ +; H_ {a，\ sigma} ^ {1}（\ mathbb {T} ^ 3））\），其中\（L_ {a，\ sigma} ^ 2 \）用齐次Sobolev–Gevrey空间\（\ dot {H} _ {a，\ sigma} ^ r \ ）当\（r = 0 \）与参数\（a \ in（0,1）\）和\（\ sigma \ ge 1 \）。我们还证明了该解在扰动下是稳定的，并且Burgers系统的长期行为由\（L_ {a，\ sigma} ^ 2 \）中的有限个自由度确定。能量方法，压紧方法和傅立叶分析是主要工具。