On self-similar singular solutions to a vorticity stretching equation

TL;DR

The paper studies a 2D nonlocal vorticity-stretching model ω_t = Z_{11} ω ω with Z_{11} = ∂_{11}Δ^{-1}. It establishes the existence of self-similar singular solutions via a spectral uncertainty principle that handles the operator's degeneracy, constructing profiles Q supported on a bounded set A satisfying Z_{11} Q = 1 on A. It also proves finite-time blow-up for compactly supported initial data with positive integral, using Fourier-analytic estimates derived from the same spectral principle. Together, these results advance understanding of singularity formation in nonlocal 2D models related to Euler dynamics and offer explicit self-similar constructions and blow-up criteria.

Abstract

We consider the following model equation: \begin{equation} ω_{t} = Z_{11}ω\,ω, \end{equation} where \begin{equation} Z_{11} = \partial_{11}Δ^{-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.

Theorems & Definitions (3)

• proof
• proof : Proof of Theorem \ref{['thm-1']}
• proof : Proof of Theorem \ref{['thm-2']}