Title: Global stability of the inhomogeneous sheared Boltzmann equation in torus Show Chinese title

Title: 环面上非均匀剪切玻尔兹曼方程的全局稳定性

Abstract: Homo-energetic solutions to the spatially homogeneous Boltzmann equation have been extensively studied, but their global stability in the inhomogeneous setting remains challenging due to unbounded energy growth under self-similar scaling and the intricate interplay between spatial dependence and nonlinear collision dynamics. In this paper, we introduce an approach for periodic spatial domains to construct global-in-time inhomogeneous solutions in a non-conservative perturbation framework, characterizing the global dynamics of growing energy. The growth of energy is shown to be governed by a long-time limit state that exhibits features not captured in either the homogeneous case or the classical Boltzmann theory. The core of our proof is the derivation of new energy estimates specific to the Maxwell molecule model. These estimates combine three key ingredients: a low-high frequency decomposition, a spectral analysis of the matrix associated with the second-order moment equation, and a crucial cancellation property in the zero-frequency mode of the nonlinear collision term. This last property bears a close analogy to the null condition in nonlinear wave equations. Show Chinese abstract

Abstract: 同能解对空间均匀玻尔兹曼方程已被广泛研究，但由于在自相似标度下能量无界增长以及空间依赖性和非线性碰撞动力学之间的复杂相互作用，在非均匀设置中的全局稳定性仍然具有挑战性。 在本文中，我们引入了一种适用于周期性空间域的方法，在非守恒扰动框架中构建全局时间的非均匀解，描述了能量增长的全局动态。 能量的增长被证明是由一个长时间极限状态所支配，该状态表现出在均匀情况或经典玻尔兹曼理论中未捕捉到的特征。 我们证明的核心是针对麦克斯韦分子模型导出新的能量估计。 这些估计结合了三个关键要素：低频-高频分解、与二阶矩方程相关的矩阵的谱分析，以及非线性碰撞项零频模式中的关键抵消性质。 这一最后性质与非线性波动方程中的零条件有密切类比。