Sharp Lower Bound for the Blow-up Rate of Solutions to the Magnetic Zakharov System without the Skin Effect

In this paper, we consider the Cauchy problem of the magnetic Zakharov system in two-dimensional space: $$\begin{array}{cccc}& \{\begin{array}{ll}& i{E}_{1t}+\mathrm{\Delta }{E}_1-n{E}_1+\eta E_2(E_1\overline{E_2}-\overline{E_1}{E}_2)=0,\\ & i{E}_{2t}+\mathrm{\Delta }{E}_2-n{E}_2+\eta E_1(\overline{E_1}{E}_2-E_1\overline{E_2})=0,\\ & n_t+\mathrm{\nabla }\cdot \mathbf{\text{v}}=0,\\ & {\mathbf{\text{v}}}_t+\mathrm{\nabla }n+\mathrm{\nabla }(\mathrm{\mid }{E}_1{\mathrm{\mid }}^2+\mathrm{\mid }{E}_2{\mathrm{\mid }}^2)=0,\end{array}& & \text{(G-Z)}\end{array}$$ with initial data $(E_{10}(x),E_{20}(x),n_0(x),{\mathbf{v}}_0(x))$, which describes the spontaneous generation of a magnetic field without the skin effect in a cold plasma, where $\eta >0$ 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 $-\mathrm{\Delta }V+V=V^3$, 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{gather*} \left|\left(E_1,E_2,n,$\mathbf{\text{v}}$\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 $\eta$ 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 $\eta$, 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.