# Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows

## TL;DR

This paper establishes the global existence and optimal decay rates of strong solutions for three-dimensional non-isentropic particle-fluid flows, using new analytical techniques for coupled kinetic-fluid systems.

## Contribution

It introduces a novel macro-micro decomposition and energy estimates to prove well-posedness and decay rates for the coupled Vlasov-Fokker-Planck and Navier-Stokes system.

## Key findings

- Global well-posedness near equilibrium for non-isentropic flows
- Algebraic decay rate in the whole space
- Exponential decay rate in periodic domain

## Abstract

In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. %The proof is based on the new macro-micro decomposition and energy estimates. New ideas and techniques are developed to establish the well-posedness and large-time behavior. For the global well-posedness our methods are based on the new macro-micro decomposition and fine energy estimates, while the proofs of the optimal large-time behavior rely on the Fourier analysis of the linearized Cauchy problem and the energy-spectrum method.