论文标题
主动矢量模型概括了3D Euler和Electron-MHD方程
Active vector models generalizing 3D Euler and electron--MHD equations
论文作者
论文摘要
我们引入了一个活跃的矢量系统，该系统概括了3D Euler方程和电子 - 磁性水动力方程（E-MHD）。我们也可以将系统视为3D Euler方程的单数化系统，在这种情况下，（E-MHD）方程对应于比3D Euler方程更单数的顺序。当未知函数沿一个坐标方向恒定时，广义的表面准晶格方程（GSQG）也可以嵌入我们系统的特殊情况中。我们研究了该系统的一些基本特性以及保护法。在系统对应于比3D Euler方程更为单数的情况下，我们在标准Sobolev空间中证明了局部适合的度。证明至关重要地取决于与Chae，Constincin，Córdoba，Gancedo和Wu的工作中使用的（GSQG）相似的急剧换向器估计。由于该系统涵盖了物理和数学上有趣的病例的许多领域，因此可以期望研究各种相关问题，其中一部分在此处讨论。
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, Córdoba, 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.
