The initial value problem for the binormal flow with rough data
The initial value problem for the binormal flow with rough data
复制标题
粗糙数据的副正态流初值问题
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
L. Vega
中科院分区：
文献类型：
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作者：
V. Banica;L. Vega
在本文中，我们考虑双法线流的初始值问题，其初始数据由规则的曲线给出，除了有拐角的一点之外。我们证明，在初始数据的适当条件下，对于严格正时间和严格负时间，存在唯一的正则解。此外，该解在所有时间都满足方程的弱版本，并且可以被视为适当选择的自相似解的扰动。相反，我们还证明，如果在时间 t = 1 时将自相似解的一个小的正则扰动作为初始条件，则存在一个唯一解，该解在时间 t = 0 时是正则的，除了在其角点与自相似解的角点角度相同之外。该解决方案可以延长负时间。该证明充分利用了之前关于自相似解的小扰动研究的论文 [9]、[2]、[3] 和 [4]。紧致性参数用于避免我们在[4]中需要的加权条件，以及对切线和法线向量的时间和空间渐近性进行更精细的分析。
In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies a weak version of the equation for all times and can be seen as a perturbation of a suitably chosen self-similar solution. Conversely, we also prove that if at time t = 1 a small regular perturbation of a self-similar solution is taken as initial condition then there exists a unique solution that at time t = 0 is regular except at a point where it has a corner with the same angle as the one of the self-similar solution. This solution can be extended for negative times. The proof uses the full strength of the previous papers [9], [2], [3] and [4] on the study of small perturbations of self-similar solutions. A compactness argument is used to avoid the weighted conditions we needed in [4], as well as a more refined analysis of the asymptotic in time and in space of the tangent and normal vectors.