Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves

TL;DR

This paper proves the global existence and convergence of solutions to the Vlasov-Poisson-Landau system towards rarefaction waves in the quasineutral limit, using weighted energy methods to handle nonlinear dynamics.

Contribution

It establishes the first rigorous derivation of the quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves, including convergence rates.

Findings

Global classical solutions exist around rarefaction waves

Solutions converge to the rarefaction wave with explicit rate

The electric potential connects fixed states at far fields

Abstract

In the paper, we consider the Cauchy problem on the spatially one-dimensional Vlasov-Poisson-Landau system modelling the motion of ions under a generalized Boltzmann relation. Let the Knudsen number and the Debye length be given as $\varepsilon>0$ and $\varepsilon^{b}$ with $\frac{3}{5}\leq b\leq 1$, respectively. As $\varepsilon\to 0$ the formal Hilbert expansion gives the fluid limit to the quasineutral compressible Euler system. We start from the small-amplitude rarefaction wave of the Euler system that admits a smooth approximation with a parameter $\delta\sim\varepsilon^{\frac{3}{5}-\frac{2}{5}a}$, where the wave strength is independent of $\varepsilon$ and we take $\frac{2}{3}\leq a\leq 1$ if $\frac{2}{3}\leq b\leq 1$ and $4-5b\leq a\leq 1$ if $\frac{3}{5}\leq b< \frac{2}{3}$. Under the scaling $(t,x)\to (\varepsilon^{-a}t,\varepsilon^{-a}x)$, for well-prepared initial data we construct the unique global classical solution to the Vlasov-Poisson-Landau system around the rarefaction wave in the vanishing limit $\varepsilon\to 0$ and also obtain the global-in-time convergence of solutions toward the rarefaction wave with rate $\varepsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln\varepsilon|$ in the $L^{\infty}_xL^2_v$ norm. The best rate is $\varepsilon^{\frac{1}{3}}|\ln\varepsilon|$ with the choice of $a=\frac{2}{3}$ and $\frac{2}{3}\leq b\leq 1$. Note that the nontrivial electric potential in the solution connects two fixed distinct states at the far fields $x=\pm\infty$ for all $t\geq 0$ and tends asymptotically as $\varepsilon\to 0$ toward a profile determined by the macro density function under the quasineutral assumption. Our strategy is based on an intricate weighted energy method capturing the quartic dissipation to give uniform bounds of the nonlinear dynamics around rarefaction waves.