超临界区域中MHD系统的强不适定性：无粘性与粘性情形
Strong ill-posedness for the MHD system in the supercritical regime: inviscid and viscous
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中文总结 AI 辅助
该研究针对三维不可压缩MHD方程组，在超临界Sobolev空间中证明了其强不适定性，引入“单磁ansatz”揭示了磁场驱动超临界不适定性的机制，填补了相关理论空白。
中文摘要 AI 辅助
本文研究三维不可压缩磁流体动力学（MHD）方程组在超临界Sobolev空间中的柯西问题。众所周知，该系统在次临界Sobolev空间中是局部适定的，而超临界区域的情况在很大程度上仍未解决。在这项工作中，我们证明了不可压缩MHD方程组在带拉普拉斯耗散和不带拉普拉斯耗散的超临界Sobolev空间中均存在范数膨胀，从而揭示了该系统在这一正则性水平下的强不适定性。我们方法的一个显著特征是引入了一种新颖的几何构造，称为“单磁 ansatz”，通过该构造，对于理想MHD系统，范数膨胀仅发生在0<s<5/2的H^s空间中的磁场b上，而速度场u的H^s范数保持一致有界。这种非对称行为表明，超临界不适定性可仅由磁场驱动，凸显了其在适定性崩溃中的关键作用。我们的发现填补了不可压缩MHD超临界正则性理论中的一个重大空白，并阐明了支配流体和磁流体动力学的不同机制。
英文摘要
This paper is concerned with the Cauchy problem for the 3D incompressible magnetohydrodynamic (MHD) equations in supercritical Sobolev spaces. It is well known that the system is locally well-posed in subcritical Sobolev spaces, whereas the supercritical regime remains largely open. In this work, we establish norm inflation for the incompressible MHD equations, both with and without Laplacian dissipation, in supercritical Sobolev spaces, thereby revealing strong ill-posedness of the system at this regularity level. A distinctive feature of our approach is the introduction of a novel geometric construction, termed the ``Magnetic-solo ansatz'', through which, for the ideal MHD system, norm inflation occurs exclusively in the magnetic field $b$ in $H^s$ with $0<s<\frac{5}{2}$, while the $H^s$-norm of the velocity field $u$ remains uniformly bounded. This asymmetric behavior shows that supercritical ill-posedness can be driven exclusively by the magnetic field, highlighting its essential role in the breakdown of well-posedness. Our findings fill a significant gap in the supercritical regularity theory for incompressible MHD and shed light on the distinct mechanisms governing the fluid and magnetic dynamics.
