发表机构
Institute for Math and AI, Wuhan University; Department of Mathematics, Shantou University（武汉大学数学与人工智能学院; 汕头大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明三维电阻MHD边界层方程在切向Sobolev空间中的线性不适定性，仅需磁场二重零点条件，通过Prandtl临界层构造与向量修正，得到指数增长解，并适用于所有切向导数损失小于1/8的情况。
AI 中文摘要
我们证明了三维电阻MHD边界层方程在切向Sobolev空间中的线性不适定性。我们假设初始切向速度分量的一个适当线性组合具有非退化临界点，并且两个初始切向磁场分量在同一点具有二重零点。与相应的二维MHD结果相比，我们的构造仅需要这个二重零点条件，而不需要磁场剪切的高阶退化。证明将三维Prandtl临界层构造与二维磁场商的向量修正相结合。该修正抵消了两个感应方程中的主导项，同时保持磁场散度约束。所得的近似解以$\exp(\sigma t/\sqrt{\varepsilon})$形式增长，其残差具有$O(\varepsilon^{-3/8})$阶的前因子，直至对数因子。然后，Duhamel论证对每个切向导数损失$\mu<\frac18$产生线性不适定性。因此，当稳定的切向磁场退化时，三维Prandtl不稳定性持续存在。
英文摘要
We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like $\exp(σt/\sqrt{\varepsilon})$, and their residuals have a prefactor of order $O(\varepsilon^{-3/8})$, up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss $μ<\frac18$. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.
