Fujita-Kato Solutions and Optimal Time Decay for the Vlasov-Navier-Stokes System in the Whole Space

TL;DR

This paper constructs global strong solutions for the 3D Vlasov-Navier-Stokes system with small initial data, extending classical results for Navier-Stokes and establishing optimal decay rates for energy and solutions.

Contribution

It extends Fujita-Kato's classical results to the Vlasov-Navier-Stokes system, introducing a higher order energy functional for decay analysis.

Findings

Global-in-time strong solutions are constructed for small initial data.

Energy decays at the optimal rate t^{-3/2} for integrable initial velocity.

In the small data case, the energy decay rate improves to t^{-5/2}.

Abstract

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical Sobolev regularity and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier-Stokes equations (NS) that has been established in 1964 by Fujita and Kato. If in addition the initial velocity is integrable, we establish that the total energy of the system decays to 0 with the optimal rate t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the use of a higher order energy functional that controls the regularity $H^1$ of the velocity and seems to have been first introduced by Li, Shou and Zhang in the context of nonhomogeneous Vlasov-Navier-Stokes system. In the small data case, we show that this energy functional decays with the rate t^{-5/2}.