论文标题
用于概括3D Euler和Electron-MHD方程的主动矢量模型的Axi对称溶液
Axi-symmetric solutions for active vector models generalizing 3D Euler and electron--MHD equations
论文作者
论文摘要
我们研究了3D不可压缩的Euler和Electron-MHD方程之间的系统，\ bekit {qore {方程*}给出 \ partial_t b + v \ cdot \ nabla b = b \ cdot \ nabla v，\ qquad v = - \ nabla \ times（-Δ） \ end {equation*}其中$ b $是$ \ mathbb {r}^3 $中的时间依赖的向量字段。假设初始数据是无旋流的公理对称性的，我们证明了Lipschitz连续溶液的局部良好性，并且在$ 1/2 <a <1 $的范围内的行驶波的存在。这些概括了3D轴对称Euler方程的相应结果，应在研究轴对称溶液的稳定性和不稳定性方面有用。
We study systems interpolating between the 3D incompressible Euler and electron--MHD equations, given by \begin{equation*} \partial_t B + V \cdot \nabla B = B\cdot \nabla V, \qquad V = -\nabla\times (-Δ)^{-a} B, \qquad \nabla\cdot B = 0, \end{equation*} where $B$ is a time-dependent vector field in $\mathbb{R}^3$. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range $1/2<a<1$. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.
