Title: Active vector models generalizing 3D Euler and electron--MHD equations Show Chinese title

Title: 活动向量模型推广三维欧拉和电子--MHD方程

Abstract: We introduce an active vector system, which generalizes both the 3D Euler equations and the electron--magnetohydrodynamic equations (E--MHD). We may as well view the system as singularized systems for the 3D Euler equations, in which case the equations of (E--MHD) correspond to the order two more singular one than the 3D Euler equations. The generalized surface quasi-geostrophic equation (gSQG) can be also embedded into a special case of our system when the unknown functions are constant in one coordinate direction. We investigate some basic properties of this system as well as the conservation laws. In the case when the system corresponds up to order one more singular than the 3D Euler equations, we prove local well-posedness in the standard Sobolev spaces. The proof crucially depends on a sharp commutator estimate similar to the one used for (gSQG) in the work of Chae, Constantin, C\'{o}rdoba, Gancedo, and Wu. Since the system covers many areas of both physically and mathematically interesting cases, one can expect that there are various related problems to be investigated, parts of which are discussed here. Show Chinese abstract

Abstract: 我们引入了一个主动向量系统，该系统同时推广了三维欧拉方程和电子-磁流体动力学方程（E-MHD）。 我们也可以将该系统视为三维欧拉方程的奇异化系统，在这种情况下，（E-MHD）的方程比三维欧拉方程更奇异两级。 当未知函数在一个坐标方向上为常数时，广义的表面准地转方程（gSQG）可以嵌入到我们系统的特殊情况中。 我们研究了该系统的某些基本性质以及守恒定律。 在系统比三维欧拉方程更奇异一级的情况下，我们在标准Sobolev空间中证明了局部适定性。 证明关键依赖于一个类似于Chae、Constantin、Córdoba、Gancedo和Wu在gSQG工作中使用的尖锐对易子估计。 由于该系统涵盖了物理和数学上许多有趣的情况，可以预期存在各种相关问题需要研究，其中一些问题在此进行了讨论。