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

TL;DR

This paper proves the existence of weak solutions for a coupled Vlasov and non-Newtonian fluid system on a periodic domain, and analyzes the large-time decay behavior depending on the fluid's stress-strain exponent.

Contribution

It establishes global weak solutions for the coupled system for all p > 8/5 and characterizes the large-time decay rates under boundedness assumptions, highlighting the influence of fluid dissipation.

Findings

Existence of weak solutions for all p > 8/5.

Algebraic decay for p > 2.

Exponential decay for 6/5 ≤ p ≤ 2.

Abstract

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.