Finite time singularity for the modified SQG patch equation

Published

Journal Article

© 016 Department of Mathematics, Princeton University. 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 α ε [0, 1/2] appearing in the kernels of their Biot-Savart laws. The values α = 0 and α = 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 suffciently small α > 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.

Full Text

Duke Authors

Cited Authors

  • Kiselev, A; Ryzhik, L; Yao, Y; ZlatoŇ°, A

Published Date

  • January 1, 2016

Published In

Volume / Issue

  • 184 / 3

Start / End Page

  • 909 - 948

International Standard Serial Number (ISSN)

  • 0003-486X

Digital Object Identifier (DOI)

  • 10.4007/annals.2016.184.3.7

Citation Source

  • Scopus