论文标题
对2D Euler方程的非对称对数螺旋涡流板解决方案的存在
Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations
论文作者
论文摘要
我们以$ m \ geq 1 $ cocentric Googarithmic螺旋形式考虑2D不可压缩的Euler方程的解决方案。我们证明了一个通用的螺旋族的存在，它们是非对称的，从某种意义上说，单个螺旋的角度并不均匀分布在单位圆上。也就是说，我们表明，如果$ m = 2 $或$ m \ geq 3 $是一个奇怪的整数，使某些非分类条件保持，那么，对于每种$ n \ in \ {1,2 \} $ M-1 $，包括亚历山大螺旋的角度的一半。我们表明，如果$ m \ in \ {2，3,5,7,9 \} $满足了非分类条件，并且所有奇数$ m> 9 $给定某个梯度矩阵的可逆，这似乎是数值计算是正确的。
We consider solutions of the 2D incompressible Euler equation in the form of $M\geq 1$ cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if $M=2$ or $M\geq 3 $ is an odd integer such that certain non-degeneracy conditions hold, then, for each $n \in \{ 1,2 \}$, there exists a logarithmic spiral with $M$ branches of relative angles arbitrarily close to $\barθ_{k} = knπ/M$ for $k=0,1,\ldots , M-1$, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if $M\in \{ 2, 3,5,7,9 \}$, and that the conditions hold for all odd $M>9$ given a certain gradient matrix is invertible, which appears to be true by numerical computations.
