论文标题
对于磁性Zakharov系统的解决方案的爆炸速率，急剧下限没有皮肤效应
Sharp Lower Bound for the Blow-up Rate of Solutions to the Magnetic Zakharov System without the Skin Effect
论文作者
论文摘要
在本文中，我们考虑了二维空间中磁性Zakharov系统的Cauchy问题：\ [\ begin {cases}＆i e_ {1t}+Δe_1-Δe_1-n e_1+ηe_2（e_1 \ ediflline {e_2}} e_ {2t}+Δe_2-n e_2+ηe_1（\ overline {e_1} e_2-e_1 \ edline {e_2}）= 0，\\＆n_t+\ nabla \ nabla \ nabla \ nabla \ cdot \ cdtbf \ cd textbf {v} = 0， （| e_1 |^2+| e_2 |^2）= 0，\\ \ \ end {cases} \ tag {g-z} \]，带有初始数据$ \ left（e_ {10}（x），e_ {20}（x），e_ {20}（x）（x），n_ {0}（0}（0}（x），x），\ mathbf {V} _} = 0} = 0}在冷等离子体中自发产生没有皮肤效应的磁场，其中$η> 0 $是物理恒定系数。冷磁场产生的两个非线性术语给古典Zakharov系统带来了不同的难度。假设初始质量满足以下估计值：\ begin {chater*} \ frac {|| q || _ {l^2（\ mathbb {r}^2）}^2} {1+η} <|| e_ {10} || _ {l^2（\ Mathbb {r}^2）}^2+|| e_ {20} || _ {l^2（\ Mathbb {r}^2）} \ end {chater*}其中$ q $是等式$-ΔV+v = v^3 $的唯一径向积极解决方案，我们证明只有一个常数$ c> 0 $，仅取决于初始数据，以至于$ t $接近$ t $ t $（爆破时间） \ left \ | \ left（e_1，e_2，n，n，\ textbf {v} \ right）\ right \ | _ {h^1（\ Mathbb {r}^2）\ times h^1（\ Mathbb {r}^2） l^2（\ mathbb {r}^2）} \ geqslant \ frac {c} {t-t-t}。 \ end {cather*}作为磁性系数$η$趋向于$ 0 $，由于merle \ cite \ cite {25frank}，爆炸率恢复了经典的2-D Zakharov系统的结果。对于任何尺寸的正$η$，在当前对初始质量的假设下，我们给出了数学上严格的理由，即在冷等离子体中存在没有皮肤效应的磁效应并不能改变最佳的下层爆炸率。
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.
