论文标题
Green的椭圆方程和3D不可压缩Euler方程的螺旋涡流贴片的结构
Structure of Green's function of elliptic equations and helical vortex patches for 3D incompressible Euler equations
论文作者
论文摘要
我们以2D有界域以差异形式开发了二阶椭圆运算符的绿色功能的新结构。基于这种结构和函数重排理论，我们将浓缩的行进旋转螺旋涡流贴片构造为无限管道中不可压缩的Euler方程。通过求解涡度的方程式\ begin {equation*} w = \ frac {1} {\ varepsilon^2} f_ \ varepsilon \ left（\ Mathcal {g} _ {g} _ {k_h} \ text {in} \ω\ end {equation*}对于小$ \ varepsilon> 0 $，并考虑了涡流的某些最大化问题，其中$ \ nathcal {g} _ {k_h} $是椭圆机$ \ \ ntercal n g n g n g n g n g n g n of in fe n g n g n g n of in l} _ =一个浓缩的螺旋涡旋斑块家族，它们渐近地趋向于由双弯曲流动进化而来的单数螺旋涡流丝。当$ p \ geq 2时，我们还在变化问题下，在变化问题下，我们还获得了最大化问题的非线性轨道稳定性。
We develop a new structure of the Green's function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity \begin{equation*} w=\frac{1}{\varepsilon^2}f_\varepsilon\left(\mathcal{G}_{K_H}w-\fracα{2}|x|^2|\ln\varepsilon|\right) \ \ \text{in}\ Ω\end{equation*} for small $ \varepsilon>0 $ and considering a certain maximization problem for the vorticity, where $ \mathcal{G}_{K_H} $ is the inverse of an elliptic operator $ \mathcal{L}_{K_H} $ in divergence form, we get the existence of a family of concentrated helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under $ L^p $ perturbation when $ p\geq 2. $
