Self-Similar Finite-Time Blowups with Smooth Profiles of the Generalized Constantin–Lax–Majda Model

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 \(a\le 1\). 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.