On stability and instability of $C^{1,\alpha}$ singular solutions to the 3D Euler and 2D Boussinesq equations

TL;DR

This paper investigates the stability and instability of $C^{1,rac{1}{2}}$ singular solutions to the 3D Euler and 2D Boussinesq equations, revealing that solutions can be both stable and unstable depending on the analytical approach used.

Contribution

It extends previous stability analyses by showing that $C^{1,rac{1}{2}}$ blowup solutions are unstable under linearized analysis but can be stable in a nonlinear, dynamically rescaled framework.

Findings

Blowup solutions are unstable under linear stability analysis.

Blowup solutions can be stable in a nonlinear, rescaled setting.

Different analytical approaches lead to contrasting stability conclusions.

Abstract

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging. In [18], Elgindi proved that the 3D axisymmetric Euler equations with no swirl and $C^{1,\alpha}$ initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with $C^{1,\alpha}$ initial velocity and boundary develop a stable asymptotically (or approximately) self-similar finite time singularity [8]. On the other hand, the authors of [35,52] recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in [35,52] require some strong regularity assumption on the initial data, which is not satisfied by the $C^{1,\alpha}$ velocity field. In this paper, we generalize the analysis of [8,18,35,52] to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with $C^{1,\alpha}$ velocity are unstable under the notion of stability introduced in [35,52]. These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. The stability analysis of the blowup solution obtained in [8,18] is based on the stability of a dynamically rescaled blowup profile in space and time, which is nonlinear in nature. The linear stability analysis in [35,52] is performed by directly linearizing the 3D Euler equations around a blowup solution in the original variables. It does not take into account the changes in the blowup time, the dynamic changes of the rescaling rate of the perturbed blowup profile and the blowup exponent of the original 3D Euler equations using a perturbed initial condition when there is an approximate self-similar blowup profile. Such information has been used in an essential way in establishing the nonlinear stability of the blowup profile in [8,18,19].