AI 中文总结
研究玻尔兹曼方程自由边界问题（模拟稀薄气体与移动壁相互作用），通过共形变换和揭示耦合能量结构，建立全耦合系统全局非线性理论，实现解的全局存在性、唯一性、稳定性及指数收敛，给出首个严格全局适定性理论。
AI 中文摘要
本文研究了玻尔兹曼方程的一个自由边界问题，该问题模拟了稀薄气体与移动壁的相互作用——这是动力学理论中的一个经典活塞问题。活塞运动由气体施加的阻力作用下的牛顿定律控制，有或没有额外的胡克恢复力。此类问题的一个主要挑战是动力学方程与自由移动边界之间的强耦合，由于解的低正则性，经典拉格朗日公式在此不可用。我们的方法依赖于两个新要素：引入共形变换将移动域简化为固定域，揭示连接动力学分布和自由边界变量的耦合能量结构以揭示界面处的内在耗散和抵消机制。这些结构观察结果导致了全耦合系统的全局非线性理论。结果，我们建立了全局存在性、唯一性、非线性稳定性以及解在全局麦克斯韦分布附近趋于平衡的指数收敛性。这为活塞型全耦合玻尔兹曼自由边界问题提供了首个严格的全局适定性理论。
英文摘要
This paper studies a free boundary problem for the Boltzmann equation that models the interaction of rarefied gas with a moving wall---a classical piston problem in kinetic theory. The piston motion is governed by Newton's law under the drag force exerted by the gas, with or without an additional Hookean restoring force. A central challenge for such a problem is the strong coupling between the kinetic equation and the free moving boundary, for which the classical Lagrangian formulation is unavailable due to low-regularity of solutions. Our approach relies on two new ingredients: a conformal transformation is introduced to reduce the moving domain to a fixed domain, and a coupled energy structure linking the kinetic distribution and free boundary variables is uncovered to reveal intrinsic dissipation and cancellation mechanisms at the interface. These structural observations lead to a global nonlinear theory for the fully coupled system. As a result, we establish the global existence, uniqueness, nonlinear stability, and exponential convergence to equilibrium of solutions near a global Maxwellian. This provides the first rigorous global well-posedness theory for a fully coupled Boltzmann free boundary problem of piston type.
Comments71 pages. All comments are welcome
