α-曲面问题自相似坍缩的数值研究
Numerical studies on the self-similar collapse of the alpha-patches problem
摘要：
本文研究了自相似变量表示的α斑块问题的动力学演化。提出了一种数值算法，并对这些方程进行了数值研究。整个文件中讨论了代码的几个基准。精确的自相似的解决方案进行了描述，并发现发挥作用，在分离崩溃从非崩溃的初始数据：小扰动，这个解决方案周围的爆炸，而其他人没有。在收敛的重新缩放剖面附近进行的数值模拟（如Cordoba等人在[12]中描述的模拟）表明，重新缩放剖面附近不存在静止图，并提出了更复杂的爆破场景。
英文摘要：
This paper studies the dynamical evolution of the alpha-patches problem expressed in self-similar variables. A numerical algorithm is proposed and these equations are numerically explored. Several benchmarks of the code are discussed throughout the paper. Exact self-similar solutions are described and are found to play a role in separating collapsing from non-collapsing initial data: small perturbations around this solution blow up while others do not. Numerical simulations performed near convergent rescaled profiles, such as those described by Cordoba et al. in [12], indicate the absence of a stationary graph in the neighborhood of the rescaled profiles and suggest a more complex scenario for blow up.
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