标题： 耦合Vlasov和非牛顿流体动力学：存在性和长时间行为 显示英文标题

标题： Coupled Vlasov and non-Newtonian fluid dynamics: existence and large-time behavior

摘要： 我们研究周期域${\mathbb T}^3$上的耦合动力学-非牛顿流体系统，其中粒子通过 Vlasov 方程演化，并通过曳力与不可压缩幂律流体相互作用。 我们证明了所有$p > \frac{8}{5}$的弱解的全局存在性，其中$p > 1$表示流体应力应变关系的幂律指数。 在粒子密度具有额外的一致有界性假设下，我们还建立了衡量速度对齐偏差的调制能量泛函的大时间衰减。 当$p > 2$时衰减率是代数的，当$\frac{6}{5} \le p \le 2$时是指数的，反映了流体耗散在长时间动力学中的作用。 显示英文摘要

摘要： We study a coupled kinetic-non-Newtonian fluid system on the periodic domain ${\mathbb T}^3$, where particles evolve by a Vlasov equation and interact with an incompressible power-law fluid through a drag force. We prove the global existence of weak solutions for all $p > \frac{8}{5}$, where $p > 1$ denotes the power-law exponent of the fluid's stress-strain relation. Under an additional uniform boundedness assumption on the particle density, we also establish large-time decay of a modulated energy functional measuring deviation from velocity alignment. The decay rate is algebraic when $p > 2$ and exponential when $\frac{6}{5} \le p \le 2$, reflecting the role of fluid dissipation in the large-time dynamics.