无趋肤效应的磁Zakharov方程组解爆破速率的精确下界
Sharp Lower Bound for the Blow-up Rate of Solutions to the Magnetic Zakharov System without the Skin Effect
摘要：
本文考虑二维Zakharov磁系统的Cauchy问题：具有初始数据$\left(E_{10}(x),E_{20}(x),n_{0}(x),\mathbf{v}_{0}(x)\right)$的\[ \begin{cases} & i E_{1t}+ΔE_1-n E_1+ηE_2 (E_1\overline{E_2}-\overline{E_1} E_2)=0, \\ & i E_{2t}+ΔE_2-n E_2+ηE_1(\overline{E_1} E_2-E_1\overline{E_2})=0, \\ & n_t+\nabla \cdot \textbf{v}=0, \\ & \textbf{v}_t+\nabla n+\nabla (|E_1|^2+|E_2|^2)=0, \\ \end{cases} \tag{G-Z} \]，它描述了冷等离子体中无趋肤效应的磁场的自发产生，其中$η>0$为物理常数系数.由冷磁场产生的两个非线性项给Zakharov系统带来了与经典系统不同的困难。假设初始质量满足以下估计：\begin{gather*} \frac{||Q||_{L^2(\mathbb{R}^2)}^2}{1+η} <||E_{10}||_{L^2(\mathbb{R}^2)}^2+||E_{20}||_{L^2(\mathbb{R}^2)}^2 <\frac{||Q||_{L^2(\mathbb{R}^2)}^2}η, \end{gather*}其中$Q$是方程$-ΔV+V=V^3 $的唯一径向正解，我们证明了存在一个常数$c>0$，它只依赖于初始数据，使得对于$t$在$T$附近（爆破时间），\begin{gather*} \left\|\left(E_1,E_2,n,\textbf{v}\right)\right\|_{H^1(\mathbb{R}^2)\times H^1(\mathbb{R}^2)\times L^2(\mathbb{R}^2)\times L^2(\mathbb{R}^2)}\geqslant \frac{c}{ T-t }. \end{gather*}由于磁系数$η$趋于$0$，爆破率恢复了Merle \cite{25Frank}的经典2-DZakharov系统的结果。对于任何大小的正$η$，在当前的假设下的初始质量，我们给出了一个数学上严格的理由的事实，即在冷等离子体中没有趋肤效应的磁效应的存在并不改变最佳的下界的爆破率。
英文摘要：
In this paper, we consider the Cauchy problem of the magnetic Zakharov system in two-dimensional space: \[ \begin{cases} & i E_{1t}+ΔE_1-n E_1+ηE_2 (E_1\overline{E_2}-\overline{E_1} E_2)=0, \\ & i E_{2t}+ΔE_2-n E_2+ηE_1(\overline{E_1} E_2-E_1\overline{E_2})=0, \\ & n_t+\nabla \cdot \textbf{v}=0, \\ & \textbf{v}_t+\nabla n+\nabla (|E_1|^2+|E_2|^2)=0, \\ \end{cases} \tag{G-Z} \] with initial data $\left(E_{10}(x),E_{20}(x),n_{0}(x),\mathbf{v}_{0}(x)\right)$, which describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, where $η>0$ is a physical constant coefficient. The two nonlinear terms generated by the cold magnetic field bring in a different difficulty from that for the classical Zakharov system. Assuming the initial mass satisfies the following estimates: \begin{gather*} \frac{||Q||_{L^2(\mathbb{R}^2)}^2}{1+η} <||E_{10}||_{L^2(\mathbb{R}^2)}^2+||E_{20}||_{L^2(\mathbb{R}^2)}^2 <\frac{||Q||_{L^2(\mathbb{R}^2)}^2}η, \end{gather*} where $Q$ is the unique radially positive solution of the equation $-ΔV+V=V^3 $, we prove that there is a constant $c>0$ depending only on the initial data such that for $t$ near $T$ (the blow-up time), \begin{gather*} \left\|\left(E_1,E_2,n,\textbf{v}\right)\right\|_{H^1(\mathbb{R}^2)\times H^1(\mathbb{R}^2)\times L^2(\mathbb{R}^2)\times L^2(\mathbb{R}^2)}\geqslant \frac{c}{ T-t }. \end{gather*} As the magnetic coefficient $η$ tends to $0$, the blow-up rate recovers the result for the classical 2-D Zakharov system due to Merle \cite{25Frank}. For any size positive $η$, under the current assumption on the initial mass, we give a mathematically rigorous justification for the fact that the presence of magnetic effects without the skin effect in the cold plasma does not change the optimal lower bound for the blow-up rates.
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