AI 中文总结
研究具有正则变化质量交换率的空间均匀玻尔兹曼方程，构造凸超线性权重，利用正则变化、质量守恒等条件，得到质量截止近似的均匀矩界，排除有限时间质量逃逸，从而获得非负初始密度下的全局积分弱解。
AI 中文摘要
我们开发了一种无凝胶化机制，该机制为具有质量交换的空间均匀玻尔兹曼方程产生全局解，适用于Grad截止硬势$B = E^\gamma b(\xi)$，其中$0 < \gamma < 1$且质量交换率呈正则变化。在不假设任何更高质量矩的情况下，我们构造了一个由二元铰链组装而成的凸超线性权重。质量相当的大粒子之间碰撞产生的加权矩通过与有界质量粒子的均匀填充库的碰撞耗散而被吸收。正则变化使这两种碰撞配置中的有符号增量可比，而质量守恒和$\gamma < 1$产生消失因子$L^{\gamma - 1}$。这在每个有限时间间隔上产生了质量截止近似的均匀矩界，并排除了有限时间内质量逃逸到无穷大的情况。因此，对于每个具有有限物理矩的非负初始密度，我们获得了一个全局积分弱解。
英文摘要
We develop a no-gelation mechanism that yields global solutions to the spatially homogeneous Boltzmann equation with mass exchange for Grad-cutoff hard potentials $B=E^γb(ξ), 0<γ<1$ and regularly varying mass-exchange rates. Without assuming any higher mass moment, we construct a convex superlinear weight assembled from dyadic hinges. Production of the weighted moment by collisions between large particles of comparable mass is absorbed by dissipation through collisions with a uniformly populated reservoir of bounded-mass particles. Regular variation makes the signed increments in these two collision configurations comparable, while mass conservation and $γ<1$ yield the vanishing factor $L^{γ-1}$. This yields a uniform moment bound of the mass-cutoff approximations on every finite time interval and rules out finite-time mass escape to infinity. Consequently, for every nonnegative initial density with finite physical moments, we obtain a global integral weak solution.
Comments40 pages, 2 figures, 1 table
