标题： 空间解析性及扩散磁流体动力学演化方程解的导数的界 显示英文标题

标题： Space analyticity and bounds for derivatives of solutions to the evolutionary equations of diffusive magnetohydrodynamics

摘要： 1981年，Foias、Guillopé和Temam对纳维-斯托克斯方程解的任意阶空间导数证明了先验估计。 这些界限在数值研究中很有启发性，通常在模拟中观察到的间歇性现象，例如Donzis等人（2013）对涡度矩的数值研究揭示了非线性的耗尽，这可能是纳维-斯托克斯方程解光滑性的原因。 我们采用一种原始方法，推导出扩散磁流体动力学演化方程的三维空间周期弱解的空间导数的类似估计。 该构造依赖于解在几乎所有时间的空间解析性。 引入了一个辅助问题，其解的Sobolev范数从下界限制了原问题解的空间解析区域的大小在$C^3$中。 我们恢复了之前对流体动力学问题获得的指数。 同样的方法也用于推导并证明弱MHD解的一阶时间导数的任意阶空间导数的类似先验边界。 本文献给乌里埃尔·弗里施教授，以纪念他80岁生日，作为对科学家和教师的感激之情。 显示英文摘要

摘要： In 1981, Foias, Guillop\'e and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier-Stokes equation. Such bounds are instructive in the numerical investigation of intermittency often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier-Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in $C^3$ of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. The same approach is also followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions. This paper is dedicated to Professor Uriel Frisch on the occasion of his 80th anniversary as a sign of appreciation of the Scientist and the Teacher.