We review recent mathematical results on the theory of ideal MHD turbulence. On the one hand, we explain a mathematical version of Taylor's conjecture on magnetic helicity conservation, both for simply and multiply connected domains. On the other hand, we describe how to prove the existence of weak solutions conserving magnetic helicity but dissipating cross helicity and energy in 3D Ideal MHD. Such solutions are bounded. In fact, we show that as soon as we are below the critical $L^3$ integrability for magnetic helicity conservation, there are weak solutions which do not preserve even magnetic helicity. These mathematical theorems rely on understanding the mathematical relaxation of MHD which is used as a model of the macroscopic behaviour of solutions of various nonlinear partial differential equations. Thus, on the one hand, we present results on the existence of weak solutions consistent with what is expected from experiments and numerical simulations, on the other hand, we show that below certain thresholds, there exist pathological solutions which should be excluded from physical grounds. It is still an outstanding open problem to find suitable admissibility conditions that are flexible enough to allow the existence of weak solutions but rigid enough to rule out physically unrealistic behaviour.

我们回顾了最近关于理想 MHD 湍流理论的数学结果。一方面，我们解释了泰勒关于磁螺旋度守恒猜想的数学版本，包括简单连通域和多重连通域。另一方面，我们描述了如何证明在 3D Ideal MHD 中存在保持磁螺旋度但耗散交叉螺旋度和能量的弱解。这样的解决方案是有界的。事实上，我们表明，只要我们低于临界值${\mathrm{大号}}^3$对于磁螺旋守恒的可积性，有一些弱解甚至不能保持磁螺旋。这些数学定理依赖于理解 MHD 的数学松弛，它被用作各种非线性偏微分方程解的宏观行为的模型。因此，一方面，我们提出了与实验和数值模拟的预期一致的弱解存在的结果，另一方面，我们表明，在某些阈值以下，存在应该从物理理由中排除的病理解. 找到合适的可接受性条件仍然是一个突出的开放性问题，这些条件足够灵活以允许弱解的存在，但又足够严格以排除物理上不切实际的行为。