Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains
Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains
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DOI：
10.1088/1361-6544/ad140c
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发表时间：
2022-07
期刊：
影响因子：
1.7
通讯作者：
D. Ambrose;P. Lushnikov;M. Siegel;Denis A. Silantyev
中科院分区：
文献类型：
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作者：
D. Ambrose;P. Lushnikov;M. Siegel;Denis A. Silantyev
考虑了具有−Λσ损耗的广义Constantin-Lax-Majda方程的整体存在性与有限时间奇点形成的问题，其中Λσ - =|k|σ，对于圆x∈[−π，π]和实直线上的问题。在周期几何中，当数据很小时，使用两种互补方法来证明σ大于或等于1的解的全局实时存在性和平流参数a的所有实值。对于不同的σ值，在a= 0和a=1/2的实线上，我们也得到了新的解析解。这些解具有自相似的有限时间奇点形成，并充分表征了奇点形成的相似指数和条件。我们重新研究了a = 0和σ = 2时实线上的Schochet解析解，并用自相似有限时间坍缩的形式重新解释了它。实线上的解析解允许任意小数据的有限时间奇点形成，即使σ值大于或等于1，从而说明实线上的问题与圆上的问题之间的关键区别。通过精确的数值模拟，可以跟踪复杂平面上奇点的形成和运动。计算验证并建立在分析理论的基础上。
The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation −Λσ , where Λσˆ=|k|σ , both for the problem on the circle x∈[−π,π] and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for σ⩾1 and all real values of an advection parameter a when the data is small. We also derive new analytical solutions in both geometries when a = 0, and on the real line when a=1/2 , for various values of σ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for a = 0 and σ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of σ that are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.