# Collapse Versus Blow-Up and Global Existence in the Generalized Constantin-Lax-Majda Equation

@article{Lushnikov2021CollapseVB, title={Collapse Versus Blow-Up and Global Existence in the Generalized Constantin-Lax-Majda Equation}, author={Pavel M. Lushnikov and Denis A. Silantyev and Michael Siegel}, journal={J. Nonlinear Sci.}, year={2021}, volume={31}, pages={82} }

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457\ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a…

## 9 Citations

### Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains

- Mathematics
- 2022

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda equation with dissipation −Λσ, where Λ̂σ = |k|σ, both for the problem…

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

- Mathematics
- 2020

It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $\omega_t + a u\omega_x = u_x \omega$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from…

### On the regularity of the De Gregorio model for the 3D Euler equations

- Mathematics
- 2021

We study the regularity of the De Gregorio (DG) model ωt + uωx = uxω on S for initial data ω0 with period π and in class X: ω0 is odd and ω0 ≤ 0 (or ω0 ≥ 0) on [0, π/2]. These sign and symmetry…

### Self-similar blow-up profile for the Boussinesq equations via a physics-informed neural network

- Mathematics, PhysicsArXiv
- 2022

A new numerical framework is developed, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations, which represents a precise description of the Luo-Hou blow-up scenario and is the first truly multi-dimensional smooth backwards selfsimilar profile found for an equation from fluid mechanics.

### Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks

- Physics, Mathematics
- 2022

We develop a new numerical framework, employing physics-informed neural networks, to ﬁnd a smooth self-similar solution for the Boussinesq equations. The solution corresponds to an asymptotic…

### On self-similar finite-time blowups of the De Gregorio model on the real line

- Mathematics
- 2022

. We show that the De Gregorio model on the real line admits inﬁnitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in ﬁnite time. These…

### Exactly self-similar blow-up of the generalized De Gregorio equation

- Mathematics
- 2022

We study exactly self-similar blow-up proﬁles fot the generalized De Gregorio model for the three-dimensional Euler equation: We show that for any α ∈ (0 , 1) such that | aα | is suﬃciently small,…

### Blow up in a periodic semilinear heat equation

- Mathematics
- 2022

Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a…

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