Optimal Lower Bound for the Blow-Up Rate of the Magnetic Zakharov System Without the Skin Effect

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Zaihui Gan, Yuchen Wang, Yue Wang, Jialing Yu
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引用次数: 0

Abstract

We focus on the following Cauchy problem of the magnetic Zakharov system in two-dimensional space:

$$\begin{aligned} \left\{ \begin{array}{ll} &{} i E_{1t}+\Delta E_1-n E_1+\eta E_2\left( E_1\overline{E_2}-\overline{E_1}E_2\right) =0, \\ &{} i E_{2t}+\Delta E_2-n E_2+\eta E_1\left( \overline{E_1}E_2-E_1\overline{E_2}\right) =0, \\ &{} n_t+\nabla \cdot {\textbf {v}}=0, \\ &{} {\textbf {v}}_t+\nabla n+\nabla \left( |E_1|^2+|E_2|^2\right) =0, \end{array} \right. \end{aligned}$$
(G-Z)
$$\begin{aligned}&(E_1,E_2,n,{\textbf {v}})(0,x)=(E_{10},E_{20},n_{0},{\textbf {v}}_{0})(x). \end{aligned}$$
(G-Z-I)

System (G–Z) describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, and \(\eta >0\) is the magnetic coefficient. The nonlinear cubic coupling terms \(E_2\left( E_1\overline{E_2}-\overline{E_1}E_2\right) \) and \(E_1\left( \overline{E_1} E_2-E_1\overline{E_2}\right) \) generated by the cold magnetic field bring  additional difficulties compared with the classical Zakharov system. For when the initial mass meets a presettable condition

$$\begin{aligned} \frac{||Q||_{L^2(\mathbb {R}^2)}^2}{1+\eta }<||E_{10}||_{L^2(\mathbb {R}^2)}^2+||E_{20}||_{L^2(\mathbb {R}^2)}^2 <\frac{||Q||_{L^2(\mathbb {R}^2)}^2}{\eta }, \end{aligned}$$

where Q is the unique radially positive solution of the equation\(-\Delta V+V=V^3 \), 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),

$$\begin{aligned} \left\| \left( E_1,E_2,n,{\textbf {v}}\right) \right\| _{H^1(\mathbb {R}^2)\times H^1(\mathbb {R}^2)\times L^2(\mathbb {R}^2)\times L^2(\mathbb {R}^2)}\geqslant \frac{c}{ T-t }. \end{aligned}$$

As the magnetic coefficient \(\eta \) tends to 0, the blow-up rate recovers the result for the classical 2-D Zakharov system due to Merle (Commun Pure Appl Math 49(8):765–794, 1996). On the other hand, for any positive \(\eta \), the result of this paper reveals a rigorous justification that the optimal lower bound of the blow-up rates is not affected by the presence of a magnetic field without the skin effect in a cold plasma.

无皮肤效应的磁性扎哈罗夫系统爆炸率的最佳下限
我们重点研究二维空间中磁扎哈罗夫系统的下列考奇问题: $$\begin{aligned}\E_1overline{E_2}-\overline{E_1}E_2\right) =0, (& {} i E_{1t}+\Delta E_1-n E_1+\eta E_2\left( E_1\overline{E_2}-\overline{E_1}E_2\right) =0, (&;{} i E_{2t}+\Delta E_2-n E_2+\eta E_1\left(\overline{E_1}E_2-E_1overline{E_2}\right) =0,\ &;{} n_t+\nabla \cdot {\textbf {v}}=0, ( &{} {\textbf {v}}_t+\nabla n+\nabla \left( |E_1|^2+|E_2|^2\right) =0, ( end{array}\right.\end{aligned}$$(G-Z)$$\begin{aligned}&(E_1,E_2,n,{\textbf {v}})(0,x)=(E_{10},E_{20},n_{0},{\textbf {v}}_{0})(x).\end{aligned}$$(G-Z-I)系统(G-Z)描述了冷等离子体中没有趋肤效应的磁场自发生成,(\ea >0\)是磁系数。与经典扎哈罗夫系统相比,冷磁场产生的非线性立方耦合项(E_2left( E_1\overline{E_2}-\overline{E_1}E_2\right) \)和(E_1left( \overline{E_1} E_2-E_1\overline{E_2}\right) \)带来了额外的困难。因为当初始质量满足一个可预设的条件时 $$\begin{aligned}\frac{||Q|||_{L^2(\mathbb {R}^2)}^2}{1+\eta }<|||E_{10}||_{L^2(\mathbb {R}^2)}^2+|||E_{20}|||_{L^2(\mathbb {R}^2)}^2 <;\其中 Q 是方程(-△ V+V=V^3 )的唯一径向正解,我们证明存在一个仅取决于初始数据的常数 \(c>0\) ,使得对于 T 附近的 t(炸毁时间),$$\begin{aligned}。\left\| \left( E_1,E_2,n,{\textbf {v}}\right) \right\| _{H^1(\mathbb {R}^2)\times H^1(\mathbb {R}^2)\times L^2(\mathbb {R}^2)\times L^2(\mathbb {R}^2)}\geqslant \frac{c}{ T-t }.\end{aligned}$$ 随着磁系数 \(\eta \)趋于 0,炸毁率恢复了梅尔(Merle)对经典二维扎哈罗夫系统的结果(Commun Pure Appl Math 49(8):765-794, 1996)。另一方面,对于任何正的(\eta \),本文的结果揭示了一个严格的理由,即炸毁率的最优下限不受冷等离子体中没有集肤效应的磁场存在的影响。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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