On a generalization of the Constantin–Lax–Majda equation

Abstract

We present evidence on global existence of solutions of De Gregorio's equation, based on numerical computation and a mathematical criterion analogous to the BealeKato-Majda theorem. Its meaning in the context of a generalized Constantin-LaxMajda equation will be discussed. We then argue that the convection term can deplete solutions of blow-up.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…The mCLM equation is part of a family of one dimensional models for the vorticity equation [37,40,107]. Its geodesic nature was recognized in [132].…”

Global Existence for the Generalized Two-Component Hunter–Saxton System

“…Another reason -and, indeed, the very incentive in [47] and here -for analyzing the family of systems (1.1), has its origin in a paradigm of Okamoto & Ohkitani [36] that the convection term can play a positive role in the global existence problem for hydrodynamically relevant evolution equations (see also [21,37]). The quadratic terms in the first component of (1.1) represent the competition in fluid convection between nonlinear steepening and amplification due to (1 − α)-dimensional stretching and κ-dimensional coupling (cf.…”

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

“…We summarize the above discussion in a Gregorio (1990), 3d Euler analogy u x = Hω ω t + uω x = u x ω global existence and regularity unknown Constantin et al (1985), analogy of vortex stretching without transport term u x = Hω ω t = u x ω finite-time blow-up from smooth data Okamoto et al (2008), a generalized model…”

“…It appears that the question of global existence of smooth solutions for smooth initial data for the 1d model (1.5) with (1.1) is open. A generalization of the model (1.4) was studied in Okamoto et al (2008) and also in Castro and Córdoba (2010), see the table below.…”