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. Poincaré C Anal. Non Linéaire, 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 ${\mathrm{\nabla }}_{x}u$, 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.