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Global existence and singularity formation for the generalized Constantin-Lax-Majda equation with dissipation: The real line vs. periodic domains by Ambrose, David M.; Lushnikov, Pavel M.; Siegel, Michael; Silantyev, Denis A. is a scholarly article available to read on EtoBox.

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

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda equation with dissipation $-\Lambda^\sigma$, where $\widehat {{\Lambda}^\sigma}=|k|^\sigma$, both for the problem on the circle $x \in [-\pi,\pi]$ and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for $\sigma \geq 1$ and all real values of an advection parameter $a$ when the data is small. We also derive new analytical solutions in both geometries when $a=0$, and on the real line when $a=1/2$, for various values of $ \sigma$. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for $a=0$ and $\sigma=2$, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of $\sigma$ that are greater than or equal to one, thereby illustrating a critical difference between the proble

Author

Ambrose, David M.; Lushnikov, Pavel M.; Siegel, Michael; Silantyev, Denis A.

Published

2022

Language

EN