Finite time singularity for the modified SQG patch equation

• 成果类型: Article
• 署名作者: Kiselev, Alexander; Ryzhik, Lenya; Yao, Yao; Zlatos, Andrej
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.3.7
• 发表日期: 2016
• 页码: 909-948
• 关键词: quasi-geostrophic equation; axisymmetrical euler equations; global well-posedness; vorticity gradient; exponential-growth; vortex patches; muskat problem; active scalar; sharp fronts; REGULARITY

摘要

It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law that is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars that interpolates between these two equations, via a parameter alpha epsilon [0, 1/2] appearing in the kernels of their Biot-Savart laws. The values alpha = 0 and alpha = 1/2 correspond to the 2D Euler and SQG cases, respectively. We prove global in time regularity for the 2D Euler patch model, even if the patches initially touch the boundary of the half-plane. On the other hand, for any sufficiently small alpha > 0, we exhibit initial data that lead to a singularity in finite time. Thus, these results show a phase transition in the behavior of solutions to these equations and provide a rigorous foundation for classifying the 2D Euler equations as critical.