On Stability and Instability of \(C^{1,\alpha }\) Singular Solutions to the 3D Euler and 2D Boussinesq Equations

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging (Majda and Bertozzi in Vorticity and incompressible flow, Cambridge University Press, Cambridge, vol 27, 2002; Gibbon in Physica D 237(14):1894–1904, 2008; Kiselev, in: Proceedings of the international congress of mathematicians, vol 3, 2018; Drivas and Elgindi in EMS Surv Math Sci 10(1):1–100, 2023; Constantin in Bull Am Math Soc 44(4):603–621, 2007). In Elgindi (Ann Math 194(3):647–727, 2021) (see also Elgindi et al. in Camb J Math 9(4), 2021), 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 self-similar (or approximately self-similar) finite time singularity (Chen and Hou in Commun Math Phys 383(3):1559–1667, 2021) in the same setting as the Hou-Luo blowup scenario (Luo and Hou in Proc Natl Acad Sci 111(36):12968–12973, 2014; Luo and Hou in SIAM Multiscale Model Simul 12(4):1722–1776, 2014). On the other hand, the authors of Vasseur and Vishik (Commun Math Phys 378(1):557–568, 2020) and Lafleche et al. (Journal de Mathématiques Pures et Appliquées 155:140–154, 2021) recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in Vasseur and Vishik (2020) and Lafleche et al. (2021) 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 Elgindi (Ann Math 194(3):647–727, 2021), Chen and Hou (Commun Math Phys 383(3):1559–1667, 2021), Vasseur and Vishik (2020) and Lafleche et al. (2021) 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 Vasseur and Vishik (2020) and Lafleche et al. (2021). These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions.