AI 中文总结
本文证明黏性非电阻不可压缩MHD方程在$\ell^1$ Besov端点空间中磁场范数在原点处膨胀，通过共振四波速度相互作用与磁场拉伸机制构造初值，表明解映射在端点拓扑下不连续。
AI 中文摘要
我们证明了在$\R^d$（$d\ge2$）上，黏性非电阻不可压缩磁流体动力学方程在原点处的磁场范数膨胀，其端点空间为\\[ \dot B^{-1}_{\infty,1}(\R^d)\times \dot B^{0}_{\infty,1}(\R^d). \\] 对于每个充分小的$\varepsilon>0$，我们构造光滑的无散度初值$(u_0,b_0)$，其中$b_0$具有紧支撑且满足\\[ \norm[\dot B^{-1}_{\infty,1}]{u_0} +\norm[\dot B^{0}_{\infty,1}]{b_0}<\varepsilon, \\] 使得相应的经典解满足\\[ \norm[\dot B^{0}_{\infty,1}]{b(t_\varepsilon)}>\varepsilon^{-1} \\] 对于某个$0<t_\varepsilon<\varepsilon$。因此，在磁场端点拓扑下，不存在与经典解一致且在原点连续的解映射。当二进求和指数大于1时可用的低-高仿积机制在$\ell^1$端点处没有增益。我们转而将共振四波速度相互作用与磁场拉伸相结合，在增长的二进壳层族上产生均匀大小的贡献。频率局部化的测试泛函提取端点下界，而加权傅里叶$L^1$空间中的一致拉格朗日估计和空间局部化论证控制余项并去除辅助常数磁场。
英文摘要
We prove magnetic-field norm inflation at the origin for the viscous, non-resistive incompressible magnetohydrodynamic equations on $\R^d$, $d\ge2$, in the endpoint space \[ \dot B^{-1}_{\infty,1}(\R^d)\times \dot B^{0}_{\infty,1}(\R^d). \] For every sufficiently small $\varepsilon>0$, we construct smooth divergence-free initial data $(u_0,b_0)$, with $b_0$ compactly supported and \[ \norm[\dot B^{-1}_{\infty,1}]{u_0} +\norm[\dot B^{0}_{\infty,1}]{b_0}<\varepsilon, \] such that the corresponding classical solution satisfies \[ \norm[\dot B^{0}_{\infty,1}]{b(t_\varepsilon)}>\varepsilon^{-1} \] for some $0<t_\varepsilon<\varepsilon$. Consequently, no solution map agreeing with classical solutions can be continuous at the origin in the magnetic endpoint topology. The low--high paraproduct mechanism available when the dyadic summability exponent is greater than one gives no gain at the $\ell^1$ endpoint. We instead combine a resonant four-wave velocity interaction with magnetic stretching to produce uniformly sized contributions on a growing family of dyadic shells. Frequency-localized test functionals extract the endpoint lower bound, while uniform Lagrangian estimates in weighted Fourier $L^1$ spaces and a spatial localization argument control the remainders and remove the auxiliary constant magnetic field.
