On self-similar singular solutions to a vorticity stretching equation

TL;DR

This paper constructs self-similar singular solutions to a vorticity stretching model involving a Calderon-Zygmond operator, demonstrating finite-time blow-up for certain initial conditions.

Contribution

It introduces a novel approach to establish the existence of self-similar singular solutions in a vorticity model with operator degeneracy, using the spectral uncertainty principle.

Findings

Existence of self-similar singular solutions.

Finite-time blow-up for compactly supported initial data with positive integral.

Overcoming operator degeneracy via spectral uncertainty principle.

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.