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arXiv 2104.05692math.APmath-phmath.MP
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弱碰撞区的Vlasov-Poisson-Landau系统

The Vlasov--Poisson--Landau system in the weakly collisional regime

Sanchit Chaturvedi, Jonathan Luk, Toan T. Nguyen

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82 pages; minor revisions and expanded introduction; version accepted for publication
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摘要：

考虑具有库仑势的Vlasov-Poisson-Landau系统在$3$ -环面上的弱碰撞区，即$$\begin{aligned} \partial_t F(t,x,v) + v_i \partial_{x_i} F(t,x,v) + E_i(t,x) \partial_{v_i} F(t,x,v) = νQ(F,F)(t,x,v),\\ E(t,x) = \nabla Δ^{-1} (\int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v - \frac{1}{(2π)^3}\int_{\mathbb T^3} \int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v \, \mathrm{d} x), \end{aligned}$$与$ν\ll 1$ .我们证明，$ε>0$足够小（但独立的$ν$），初始数据是$O(εν^{1/3})$ -索伯列夫空间扰动从全球麦克斯韦导致全球的时间解决方案收敛到全球麦克斯韦作为$t\to \infty$。该解决方案表现出均匀的$ν$朗道阻尼和增强的耗散。我们的主要结果是类似的Bedrossian的Vlasov-Poisson-Fokker-Planck方程具有相同的阈值较早的结果。然而，与Fokker-Planck情况不同，由于朗道碰撞算子的复杂性，线性算子不能被显式反演。为此，我们发展了一个基于能量的框架，它结合了郭的加权能量方法与次矫顽能方法和交换向量场方法。证明也依赖于线性化密度方程的逐点预解估计。

英文摘要：

Consider the Vlasov-Poisson-Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $$\begin{aligned} \partial_t F(t,x,v) + v_i \partial_{x_i} F(t,x,v) + E_i(t,x) \partial_{v_i} F(t,x,v) = νQ(F,F)(t,x,v),\\ E(t,x) = \nabla Δ^{-1} (\int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v - \frac{1}{(2π)^3}\int_{\mathbb T^3} \int_{\mathbb R^3} F(t,x,v)\, \mathrm{d} v \, \mathrm{d} x), \end{aligned}$$ with $ν\ll 1$. We prove that for $ε>0$ sufficiently small (but independent of $ν$), initial data which are $O(εν^{1/3})$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to \infty$. The solutions exhibit uniform-in-$ν$ Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov-Poisson-Fokker-Planck equation with the same threshold. However, unlike in the Fokker-Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo's weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation.

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