论文标题
3D磁流体动力方程的较弱溶液的急剧非唯一性
Sharp non-uniqueness of weak solutions to 3D magnetohydrodynamic equations
论文作者
论文摘要
我们证明了在类别$ l^γ_tw^{s，p} _x $的类中，弱解决方案对3D超级粘性和电阻MHD的非唯一性，其中指数$（s，s，γ，p）$在两个超临界方案中。结果表明，即使在狮子指数以外的高度粘性和电阻性方面，也是检测非唯一性的正确标准。特别是，对于经典的粘性和电阻性MHD，非唯一性在LPS条件的端点$（0,2，\ Infty）附近是尖锐的。此外，构造的弱解决方案承认，在及时设置的小分形数字之外的部分规律性，零$ \ mathcal {h}^{η_*} $ - hausdorff dimension，其中$η_*$可以是任何给定的小正常。此外，我们证明了强烈的消失粘度和电阻率结果，这会导致泰勒的猜想沿弱溶液的某些子序列失败，以超过狮子指数以外的超粘性和电阻性MHD。
We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^γ_tW^{s,p}_x$, where the exponents $(s,γ,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Ladyženskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regularity outside a small fractal singular set in time with zero $\mathcal{H}^{η_*}$-Hausdorff dimension, where $η_*$ can be any given small positive constant. Furthermore, we prove the strong vanishing viscosity and resistivity result, which yields the failure of Taylor's conjecture along some subsequence of weak solutions to the hyper viscous and resistive MHD beyond the Lions exponent.
