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

TL;DR

This paper establishes a sharp lower bound on the blow-up rate of solutions to the magnetic Zakharov system in 2D without the skin effect, showing it matches the classical Zakharov system as the magnetic coefficient tends to zero.

Contribution

It provides the first rigorous lower bound for blow-up rates in the magnetic Zakharov system without skin effect, extending classical results to include magnetic effects.

Findings

Lower bound of 1/(T-t) for blow-up rate proven

Magnetic effects do not alter the optimal blow-up rate

As magnetic coefficient approaches zero, results recover classical Zakharov system behavior

Abstract

In this paper, we consider the Cauchy problem of the magnetic Zakharov system in two-dimensional space: \[ \begin{cases} & i E_{1t}+\Delta E_1-n E_1+\eta E_2 (E_1\overline{E_2}-\overline{E_1} E_2)=0, \\ & i E_{2t}+\Delta E_2-n E_2+\eta E_1(\overline{E_1} E_2-E_1\overline{E_2})=0, \\ & n_t+\nabla \cdot \textbf{v}=0, \\ & \textbf{v}_t+\nabla n+\nabla (|E_1|^2+|E_2|^2)=0, \\ \end{cases} \tag{G-Z} \] 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 $\eta>0$ 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+\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{gather*} 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{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 $\eta$ 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.