Self-binormal solutions of the Localized Induction Approximation: Singularity formation

On self-similar singularity formation for the binormal flow

The aim of this article is to establish a concise proof for a stability result of self-similar solutions of the binormal flow, in some more restrictive cases than in [5]. This equation, also known as the Local Induction Approximation, is a standard model for vortex filament dynamics, and its self-similar solution describes the formation of a corner singularity on the filament. Our approach strongly uses the link that Hasimoto pointed out in 1972 between the solution of the binormal flow and the one of the 1-D cubic Schrödinger equation, as well as the existence results associated to the latter…

On the energy of critical solutions of the binormal flow

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schrödinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conserv…

Stability of the selfsimilar dynamics of a vortex filament

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schrödinger equation, connected to the binormal flow by Hasimoto's transfo…

Scattering for 1D cubic NLS and singular vortex dynamics

In this paper we study the stability of the self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions $χ_a(t,x)$ form a family of evolving regular curves of $\mathbb R^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of $χ_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence for the binormal flow. Nevertheless, w…

粘性涡旋细丝的演化与孤子型传播

Evolution of viscous vortex filaments and soliton-type propagation

研究粘性不可压缩流体初始涡度在光滑曲线时的演化，通过特定构造及应用于桥本孤子，得出涡度主导阶描述及相关结论，包括动能分布局部部分在移动区域的特性等。

Schroedinger Map自相似解的数值研究

A numerical study of the self-similar solutions of the Schroedinger Map

通过数值方法研究涡丝局部感应近似自相似解（单参数族）的有限时间奇点形成机制，并探讨相关边界条件。