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

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.