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

We focus on the following Cauchy problem of the magnetic Zakharov system in two-dimensional space: G-Z \(\begin{aligned} \left\{ \begin{array}{ll} &{} i E_{1t}+\Delta E_1-n E_1+\eta E_2\left( E_1\overline{E_2}-\overline{E_1}E_2\right) =0, \\ &{} i E_{2t}+\Delta E_2-n E_2+\eta E_1\left( \overline{E_1}E_2-E_1\overline{E_2}\right) =0, \\ &{} n_t+\nabla \cdot {\textbf {v}}=0, \\ &{} {\textbf {v}}_t+\nabla n+\nabla \left( |E_1|^2+|E_2|^2\right) =0, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& i{E}_{1t}+\mathrm{\Delta }{E}_1-n{E}_1+\eta E_2\left(E_1,\overline{E_2},-,\overline{E_1},E_2\right)=0,\hfill \\ \hfill & i{E}_{2t}+\mathrm{\Delta }{E}_2-n{E}_2+\eta E_1\left(\overline{E_1},E_2,-,E_1,\overline{E_2}\right)=0,\hfill \\ \hfill & n_t+\mathrm{\nabla }·\mathbf{v}=0,\hfill \\ \hfill & {\mathbf{v}}_t+\mathrm{\nabla }n+\mathrm{\nabla }\left(|,E_1,{|}^2,+,{|E_2|}^2\right)=0,\hfill \end{array}\right)\end{array}$$ G-Z-I \(\begin{aligned}&(E_1,E_2,n,{\textbf {v}})(0,x)=(E_{10},E_{20},n_{0},{\textbf {v}}_{0})(x). \end{aligned}\) $$\begin{array}{cc}& (E_1,E_2,n,\mathbf{v})(0,x)=(E_{10},E_{20},n_0,{\mathbf{v}}_0)(x).\hfill \end{array}$$ System (G–Z) describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, and \(\eta >0\) $\eta >0$ is the magnetic coefficient. The nonlinear cubic coupling terms  \(E_2\left( E_1\overline{E_2}-\overline{E_1}E_2\right) \) $E_2\left(E_1,\overline{E_2},-,\overline{E_1},E_2\right)$ and \(E_1\left( \overline{E_1} E_2-E_1\overline{E_2}\right) \) $E_1\left(\overline{E_1},E_2,-,E_1,\overline{E_2}\right)$ generated by the cold magnetic field bring  additional difficulties compared with the classical Zakharov system. For when the initial mass meets a presettable condition \(\begin{aligned} \frac{||Q||_{L^2(\mathbb {R}^2)}^2}{1+\eta }<||E_{10}||_{L^2(\mathbb {R}^2)}^2+||E_{20}||_{L^2(\mathbb {R}^2)}^2 <\frac{||Q||_{L^2(\mathbb {R}^2)}^2}{\eta }, \end{aligned}\) $$\begin{array}{c}\hfill \frac{{||Q||}_{L^2({\mathbb{R}}^2)}^2}{1+\eta }<||E_{10}{||}_{L^2({\mathbb{R}}^2)}^2+||E_{20}{||}_{L^2({\mathbb{R}}^2)}^2<\frac{{||Q||}_{L^2({\mathbb{R}}^2)}^2}{\eta },\end{array}$$ where Q is the unique radially positive solution of the equation \(-\Delta V+V=V^3 \) $-\mathrm{\Delta }V+V=V^3$ , we prove that there is a constant  \(c>0\) $c>0$   depending only on the initial data such that for t near T (the blow-up time), \(\begin{aligned} \left\| \left( E_1,E_2,n,{\textbf {v}}\right) \right\| _{H^1(\mathbb {R}^2)\times H^1(\mathbb {R}^2)\times L^2(\mathbb {R}^2)\times L^2(\mathbb {R}^2)}\geqslant \frac{c}{ T-t }. \end{aligned}\) $$\begin{array}{c}\hfill {∥\left(E_1,,,E_2,,,n,,,\mathbf{v}\right)∥}_{H^1({\mathbb{R}}^2)\times H^1({\mathbb{R}}^2)\times L^2({\mathbb{R}}^2)\times L^2({\mathbb{R}}^2)}⩾\frac{c}{T-t}.\end{array}$$ As the magnetic coefficient \(\eta \) $\eta$ 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 \(\eta \) $\eta$ , 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.