| Abstract: |
| The question of global existence versus finite-time singularity formation is considered for the generalized Constantin-Lax-Majda (gCLM) equation with dissipation. This equation was first introduced by Constantin, Lax and Majda as a simplified model for singularity formation in the 3D incompressible Euler equations. It was later generalized to include dissipation and 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 methods to study singularity formation in the 3D Euler equations. Despite significant effort, little is known about global existence versus singularity formation for general values of $a$. We use two complementary approaches to prove global-in-time existence of solutions for small data in the periodic geometry, when dissipation is strong enough. We also find a significant difference between the problems in the periodic and real-line geometries when dissipation is present. |
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