On the Slightly Perturbed De Gregorio Model on $S^1$

Abstract

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) ${\omega }_t+au{\omega }_x=u_x\omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a<1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $||\omega (t)|{|}_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.

arXiv comment19 pages. Added discussions in the introduction

References in corpus (9)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• On the stability of the notion of non-characteristic point and blow-up profile for semilinear wave equations
• Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha }$ velocity and boundary
• On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equation
• On the Finite-Time Blowup of a 1D Model for the 3D Axisymmetric Euler Equations
• On the De Gregorio modification of the Constantin-Lax-Majda Model
• Singularity Formation and Global Well-Posedness for the Generalized Constantin-Lax-Majda Equation with Dissipation
• Collapse vs. blow up and global existence in the generalized Constantin-Lax-Majda equation
• Finite-Time Singularity Formation for $C^{1,\alpha }$ Solutions to the Incompressible Euler Equations on ${\mathbb{R}}^3$

Cited by in corpus (4)

• Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model
• On self-similar finite-time blowups of the De Gregorio model on the real line
• 1D Model for the 3D Magnetohydrodynamics
• On the regularity of the De Gregorio model for the 3D Euler equations