Global existence and time decay of strong solutions to a fluid-particle coupled model with energy exchanges

TL;DR

This paper proves the global existence and optimal decay rates of strong solutions for a 3D fluid-particle coupled model involving Navier-Stokes and Vlasov-Fokker-Planck equations, using energy and Fourier analysis methods.

Contribution

It establishes the first rigorous results on global existence and decay rates for this coupled fluid-particle model with energy exchanges in three-dimensional space.

Findings

Solutions decay as (1+t)^{-3/4} in L^2-norm.

Gradients decay as (1+t)^{-5/4}.

Highest-order derivatives decay as (1+t)^{-7/4} in L^2.

Abstract

In this paper, we investigate a three-dimensional fluid-particle coupled model. % in whole space $\mathbb{R}^3$. This model combines the full compressible Navier-Stokes equations with the Vlasov-Fokker-Planck equation via the momentum and energy exchanges. We obtain the global existence and optimal time decay rates of strong solutions to the model in whole space $\mathbb{R}^3$ when the initial data are a small perturbation of the given equilibrium in $H^2$. We show that the $L^2$-norms of the solutions and their gradients decay as $(1+t)^{-3/4}$ and $(1+t)^{-5/4}$ respectively. Moreover, we also obtain the decay rates of solutions in $L^p$-norms for $p\in [2,\infty]$, and the optimal time decay rates of the highest-order derivatives of strong solutions which reads as $(1+t)^{-{7}/{4}}$ in $L^2$-norm. % Our decay rates are consistent with those of non-isentropic compressible Navier-Stokes equations. When the model is considered in a periodic domain, besides the global existence results, we show the strong solution decay exponentially. Our proofs rely on the energy method, Fourier analysis techniques, and the method of frequency decomposition. And some new ideas are introduced to achieve the desired convergence rates.