论文标题
小型数据解决方案的大量时间行为在整个空间
Large time behavior of small data solutions to the Vlasov-Navier-Stokes system on the whole space
论文作者
论文摘要
我们研究了$ \ r^3 \ times \ r^3 $的小型数据解决方案的较大时间行为。我们证明，动力学分布函数集中于$ 0 $支撑的Dirac质量，而流体速度均匀均以$ 0 $，均以多项式速率均以$ 0 $。证明是基于两个步骤的，遵循\ cite {hkmm}中规定的一般策略：（1）假设对动力学密度的统一控制，系统的能量会以多项式速率衰减，（2）引导程序参数允许获得这样的控制。最后一步需要对所谓的布林克曼力量的结构有充分的了解，这是从与Vlasov-Navier-Stokes System相关的耗散（以及更高版本）的新身份家族中进行的。
We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system on $\R^3 \times \R^3$. We prove that the kinetic distribution function concentrates in velocity to a Dirac mass supported at $0$, while the fluid velocity homogenizes to $0$, both at a polynomial rate. The proof is based on two steps, following the general strategy laid out in \cite{HKMM}: (1) the energy of the system decays with polynomial rate, assuming a uniform control of the kinetic density, (2) a bootstrap argument allows to obtain such a control. This last step requires a fine understanding of the structure of the so-called Brinkman force, which follows from a family of new identities for the dissipation (and higher versions of it) associated to the Vlasov-Navier-Stokes system.
