Global Existence and Large Time Behavior of Classical Solutions to the Incompressible Inhomogeneous Kinetic-Fluid Model With Energy Exchanges

In this paper, we concentrate on a kinetic-fluid model with energy exchanges, which contains a Vlasov–Fokker–Planck equation for the particles and an incompressible inhomogeneous Navier–Stokes system with heat conductivity for the fluid. The two parts in the model twist together via the momentum and energy exchanges revealing the interaction between the fluid and the particles. For the small initial data near the given equilibrium state, we obtain the global existence, uniqueness, and optimal decay rates of classical solutions in the three-dimensional whole space. Furthermore, we obtain the optimal decay rates of the gradients in $x$ of solutions. For the periodic domain case, the decay rates are exponential. The proofs of our results are mainly based on the new macro–micro decomposition and modified energy-spectrum method. We also introduce some new ideas and develop a new energy method to enclose the a priori estimates. More precisely, we first exploit the new macro–micro decomposition and the dissipation rate on the gradient of the pressure $p$ instead of the density $\rho$ in $H^2$ norm to enclose the estimates ${\parallel f\parallel }_{H_{x,\xi }^3}+{\parallel u\parallel }_{H_x^3}$. Then, by applying Gronwall's inequality and establishing the ${(1+t)}^{-5/4}$ decay rate in the whole space and the exponential decay rate on the periodic domain of ${\parallel \nabla u\parallel }_{H_x^2}$, we enclose the estimate ${\parallel \rho \parallel }_{H_x^3}$ in the a priori estimates. Finally, based on the modified energy-spectrum method, and the fact that the transport equation with divergence free velocity field is $L^1$ conserved, we obtain the optimal decay rates of the solutions $(f,u,\theta )$ and their gradients.