Gevrey regularity and analyticity for the solutions of the Vlasov-Navier-Stokes system

In this paper, we prove propagation of $\frac{1}{s}$-Gevrey regularity $(s \in (0, 1))$ and analyticity $(s=1)$ for the Vlasov-Navier-Stokes system on $\mathbb{T}^d \times \mathbb{R}^d$ (and $\mathbb{R}^d\times\mathbb{R}^d$) using a Fourier space method in analogy to the results proved for the Euler system in [Kukavica and Vicol, Proc. Amer. Math. Soc., 2009] and [Levermore and Oliver, JDE, 1997] and for Vlasov-Poisson system in [Velozo Ruiz, Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire, 2021]. More precisely, we give quantitative estimates for the growth of the $\frac{1}{s}$-Gevrey norm and decay of the regularity radius for the solution of the system in terms of $\nabla_x u$, the spatial density $\rho_f$ and the diameter of the support in the velocity variable of the distribution of particles $f$. In particular, this implies existence of $\frac{1}{s}$-Gevrey $(s \in (0, 1))$ and analytic $(s = 1)$ solutions for the Vlasov-Navier-Stokes system in $\mathbb{T}^d\times\mathbb{R}^d$ (and $\mathbb{R}^d\times\mathbb{R}^d$), and global Gevrey solutions in $\mathbb{T}^3\times\mathbb{R}^3$ for sufficiently small data, and an initial data for the Vlasov equation with compact support in velocity.