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

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.