论文标题
vlasov-poisson-landau系统的全球准中性欧拉限制
Global quasineutral Euler limit for the Vlasov-Poisson-Landau system with rarefaction waves
论文作者
论文摘要
在本文中，我们考虑了在空间上的一维vlasov-Poisson-Landau系统上的库奇问题，该系统在广义玻尔兹曼关系下对离子的运动进行了建模。让Knudsen编号和Debye长度分别以$ \ VAREPSILON> 0 $和$ \ VAREPSILON^{B} $，分别为$ \ frac {3} {5} \ leq b \ leq 1 $。作为$ \ varepsilon \ to 0 $，正式的希尔伯特扩展给了准压缩欧拉系统的流体限制。我们从Euler系统的小振幅稀有浪潮开始，该浪潮以参数$δ\ sim \ sim \ varepsilon^{\ frac {\ frac {3} {5} {5} {5} - \ frac {2} {2} {5} {5} a} $独立于$ \ varepsilon $ and weepsilon $ and， $ \ frac {2} {3} \ leq a \ leq 1 $如果$ \ frac {2} {3} {3} \ leq b \ leq 1 $和$ 4-5B \ leq a \ leq a \ leq 1 $ $ 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带速率的稀有波$ \ varepsilon^{\ frac {\ frac {3} {5} - \ frac {2} {5} {5} a} | \ ln \ ln \ varepsilon | $ in $ l^{\ infty} __xl^_xl^2_v $ norm。最佳费率是$ \ varepsilon^{\ frac {1} {3}} | \ ln \ varepsilon | $，选择$ a = \ frac {2} {3} {3} $和$ \ frac {2} {2} {3} {3} {3} {3} \ leq b \ \ leq leq 1 $。请注意，解决方案中的非平凡电势将所有$ x = \ pm \ pm \ geq 0 $ $ x = \ pm \ geq 0 $ $ dyt the undy $ bysymptototty上趋向于$ \ varepsilon \ 0 $趋于由宏密度函数确定的配置文件。我们的策略基于一种复杂的加权方法，该方法捕获了四分之一的耗散，从而在稀疏波周围产生了非线性动力学的均匀界限。
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 $δ\sim\varepsilon^{\frac{3}{5}-\frac{2}{5}a}$, 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.
