Exact periodic solutions of the generalized Constantin-Lax-Majda equation with dissipation

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.