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arXiv · 2305.05895

Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model

De Huang·Xiang Qin·Xiuyuan Wang·Dongyi Wei
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Abstract

We show that the $a$-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an $a$-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of $a$ and also explains previous numerical observations for a wide range of $a$.

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De Huang, Xiang Qin, Xiuyuan Wang, Dongyi Wei. 2023-05-10. Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model. https://doi.org/10.1007/s00205-024-01971-3

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