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Exact Periodic Solutions of the Generalized Constantin–Lax–Majda Equation With Dissipation
Journal article   Peer reviewed

Exact Periodic Solutions of the Generalized Constantin–Lax–Majda Equation With Dissipation

Denis A. Silantyev, Pavel M. Lushnikov, Michael Siegel and David M. Ambrose
Studies in applied mathematics (Cambridge), v 155(3), e70115
Sep 2025
DOI: https://doi.org/10.1111/sapm.70115

Abstract

complex singularities global existence pole dynamics pole solutions self‐similar finite‐time singularity formation Fluid Dynamics
ABSTRACT We present exact pole dynamics solutions to the generalized Constantin–Lax–Majda (gCLM) equation in a periodic geometry with dissipation −Λσ$-\Lambda ^\sigma$, where its spatial Fourier transform is Λσ̂=|k|σ$\widehat{\Lambda ^\sigma }=|k|^\sigma$. The gCLM equation is a simplified model for singularity formation in the 3D incompressible Euler equations. It includes an advection term with parameter a$a$, 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$a=0$ and 1/2$1/2$ and σ=0$\sigma =0$ and 1, for which a closed collection of (periodically repeated) poles evolve in the complex plane. Self‐similar finite‐time blowup of the solutions is analyzed and compared for the different values of σ$\sigma$, 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$a=0$, σ≥0$\sigma \ge 0$. Several interesting features of the solutions are discussed.

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Collaboration types
Domestic collaboration
Web of Science research areas
Mathematics, Applied
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