论文标题
不可压缩的欧拉方程中的行进螺旋螺旋和涡旋细丝猜想
Travelling helices and the vortex filament conjecture in the incompressible Euler equations
论文作者
论文摘要
我们考虑以涡旋形式表示的$ {\ mathbb r}^3 $中的欧拉方程。可以追溯到赫尔姆霍尔茨的一个经典问题是描述曲线周围高浓度的溶液的演变。 Da Rios在1906年的工作指出，这种曲线必须通过所谓的二手曲率流进行演变。真正的解决方案的存在集中在该法律发展的给定曲线附近，这是一个长期存在的开放问题，仅针对沿轴持续速度持续速度的特殊情况回答了薄涡旋环。我们提供了与翻译旋转螺旋相关的{\ em螺旋细丝}的第一个严格结构。该解决方案始终定义，并且不会随时间变化。结果概括为多个类似的螺旋丝一起行进和旋转。
We consider the Euler equations in ${\mathbb R}^3$ expressed in vorticity form. A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of {\em helical filaments}, associated to a translating-rotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple similar helical filaments travelling and rotating together.
