On self-similar singular solutions to a vorticity stretching equation

Abstract

We consider the following model equation: \begin{equation} \omega_{t} = Z_{11}\omega\,\omega , \end{equation} where \begin{equation} Z_{11} = \partial_{11}\Delta^{-1} \end{equation} is a Calderon-Zygmond operator. We get the existence of self-similar singular solutions with a special form. The main difficulty is the degeneracy of the operator $Z_{11}$ that is overcome by the spectral uncertainty principle. We also show that the solution to this model blows up in finite time if the initial datum is compactly supported and has a positive integral.