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Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation
04 Nov 2024
Abstract
We present exact pole dynamics solutions to the generalized
Constantin-Lax-Majda (gCLM) equation in a periodic geometry with dissipation
$-\Lambda^\sigma$, where its spatial Fourier transform is
$\widehat{\Lambda^\sigma}=|k|^\sigma$. The gCLM equation is a simplified model
for singularity formation in the 3D incompressible Euler equations. It includes
an advection term with parameter $a$, which allows different relative weights
for advection and vortex stretching. There has been intense interest in the
gCLM equation, and it has served as a proving ground for the development of
methods to study singularity formation in the 3D Euler equations. Several exact
solutions for the problem on the real line have been previously found by the
method of pole dynamics, but only one such solution has been reported for the
periodic geometry. We derive new periodic solutions for $a=0$ and $1/2$ and
$\sigma=0$ and $1$, for which a closed collection of (periodically repeated)
poles evolve in the complex plane. Self-similar finite-time blow-up of the
solutions is analyzed and compared for the different values of $\sigma$, and to
a global-in-time well-posedness theory for solutions with small data presented
in a previous paper of the authors. Motivated by the exact solutions, the
well-posedness theory is extended to include the case $a=0$, $\sigma \geq 0$.
Several interesting features of the solutions are discussed.
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Details
- Title
- Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation
- Creators
- Denis A SilantyevPavel M LushnikovMichael SiegelDavid M Ambrose
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021955910804721