We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics (MHD) in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the L³ integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for ideal MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.

中文翻译：

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磁螺旋度、弱解和理想 MHD 的弛豫

我们在弱解的背景下重新审视磁螺旋性守恒问题和磁流体动力学 (MHD) 中的 Woltjer-Taylor 弛豫理论。我们为理想 MHD 系统引入了一个松弛系统，该系统以与等离子体湍流中观察到的一致的方式解耦了流体动力湍流的影响，例如磁螺旋度守恒中雷诺应力项的出现。作为副产品，我们回答了该领域的两个悬而未决的问题：我们展示了磁螺旋度守恒的L ³可积条件的清晰度，并为理想的 MHD 耗散能量和交叉螺旋度提供了湍流有界解，但具有（任意）恒定的磁螺旋度。