Formation of small scales in nonlinear PDEs

Small scale creation in solutions to modified SQG equations

Alex Kiselev

Duke University

The modified SQG equations have similarities both to 2D and 3D Euler equations.The question of global regularity vs finite time blow up remains open for smooth initial data, for the entire range of Biot-Savart law parameter 0<\alpha<1. For patch solutions in half-plane, an example of finite time singularity has been constructed by Ryzhik, Yao, Zlatos and myself in the case of small $\alpha$. However, for smooth initial data there have been no examples of any infinite in time growth. In this talk I will review the history and describe a recent construction by Siming He and myself yielding solutions with exponential in time growth of derivatives.