Global well-posedness and decay rates of strong solutions to the incompressible Vlasov-MHD system

TL;DR

This paper establishes the global existence and decay rates of strong solutions to an incompressible Vlasov-MHD system, overcoming key mathematical challenges with innovative methods, and provides the first such results for this nonlinear plasma model.

Contribution

It introduces the first rigorous proof of global well-posedness and decay rates for strong solutions to the nonlinear Vlasov-MHD system with Lorentz forces.

Findings

Global solutions exist under small initial data assumptions.

Solutions decay polynomially in the whole space and exponentially on the torus.

The paper employs characteristic methods and Fourier analysis to overcome mathematical difficulties.

Abstract

In this paper, we study the global well-posedness and decay rates of strong solutions to an incompressible Vlasov-MHD model arising in magnetized plasmas. This model is consist of the Vlasov equation and the incompressible magnetohydrodynamic equations which interacts together via the Lorentz forces. It is readily to verify that it has two equilibria $(\bar f,\bar u,\bar B)=(0,0,0)$ and $( \tilde f,\tilde u,\tilde B)=(M,0,0)$, where $M$ is the global maxwellian. For each equilibrium, assuming that the $H^2$ norm of the initial data $(f_0,B_0,U_0)$ is sufficient small and $f_0(x,v)$ has a compact support in the position $x$ and the velocity $v$, we construct the global well-posedness and decay rates of strong solutions near the equilibrium in the whole space $\mathbb{R}^3$. And the solution decays polynomially. The global existence result still holds for the torus $\mathbb{T}^3$ case without the compact support assumption in $x$. In addition, the decay rates are exponential. Lack of dissipation structure in the Vlasov equation and the strong trilinear coupling term $((u-v)\times B)f$ in the model are two main impediments in obtaining our results. To surround these difficulties, we assume that $f_0(x,v)$ has a compact support and utilize the method of characteristics to calculate the size of the supports of $f$. Thus, we overcome the difficulty in estimating the integration $\int_{\mathbb{R}^3} \big((u-v)\times B\big)f\mathrm{d}v$ and obtain the global existence of strong solutions by taking advantage of a refined energy method. Moreover, by making full use of the Fourier techniques, we obtain the optimal time decay rate of the gradient of the solutions. This is the first result on strong solutions to the Vlasov-MHD model containing nonlinear Lorentz forces.