发表机构
Tsinghua University; National Center for Theoretical Sciences, National Taiwan University; Pohang University of Science and Technology（清华大学; 台湾大学理论科学中心; 浦项科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对三维环面带硬势的截断玻尔兹曼方程，建立非微扰全局动力学条件理论，提出尾到强制性原理，获低正则性替代方案，为带物理边界条件的相关问题提供研究路径。
AI 中文摘要
我们针对三维环面上带硬势的截断玻尔兹曼方程，建立了真正非微扰全局动力学的条件理论。我们证明，对于合适的拉伸指数权重W，满足\\( \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\\,dv<\infty \\)的任意全局弱解，都具有一致加权\\( L^\infty \\)控制，在耗散弱类中是唯一的，且指数收敛到全局麦克斯韦分布。特别地，对于具有一致拉伸指数速度上界的任意振幅解，这些结论均成立。关键机制是“尾到强制性原理”：一致速度尾控制无需先验施加宏观强制性，即可产生局部密度和非线性碰撞频率的正下界。双杜哈梅尔正性论证还能在任意短的正时间产生高斯下界，允许存在含局部真空的初始数据。我们在额外空间均匀速度下轮廓的条件下，建立了多项式加权对应理论，得到了类似的加权\\( L^\infty \\)界、唯一性和指数弛豫。我们的结果为需要光滑性和逐点宏观界的条件理论提供了低正则性替代方案，揭示了单一速度尾条件如何控制大数据玻尔兹曼动力学，并为带物理边界条件的问题指明了方向。
英文摘要
We establish a conditional theory of genuinely non-perturbative global dynamics for the cutoff Boltzmann equation with hard potentials on the three-dimensional torus. We prove that any global mild solution satisfying \[ \sup_{t\ge0}\int_{\mathbb R^3}W(v)\sup_{x\in\mathbb T^3}F(t,x,v)\,dv<\infty, \] for a suitable stretched-exponential weight \(W\), enjoys uniform weighted \(L^\infty\) control, is unique in the dissipative mild class, and converges exponentially to the global Maxwellian. In particular, these conclusions hold for arbitrary-amplitude solutions admitting a uniform stretched-exponential velocity upper bound. The key mechanism is a *tail-to-coercivity principle*: uniform velocity-tail control generates positive lower bounds for the local density and nonlinear collision frequency, without imposing macroscopic coercivity a priori. A double-Duhamel positivity argument also generates a Gaussian lower bound at arbitrarily short positive times, allowing initial data with local vacuum. We establish a polynomially weighted counterpart under an additional spatially uniform lower velocity profile, obtaining analogous weighted \(L^\infty\) bounds, uniqueness, and exponential relaxation. Our results provide a low-regularity alternative to conditional theories requiring smoothness and pointwise macroscopic bounds. They reveal how a single velocity-tail condition controls large-data Boltzmann dynamics and suggest a route toward problems with physical boundary conditions.
Comments119 pages
