On stability and instability of \(C^{1, \alpha}\) singular solutions to the 3D Euler and 2D Boussinesq equations. (English) Zbl 1541.35359
Summary: Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging
[A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow. Cambridge: Cambridge University Press (2002; Zbl 0983.76001);
J. D. Gibbon, Physica D 237, No. 14–17, 1894–1904 (2008; Zbl 1143.76389);
A. Kiselev, in: Proceedings of the international congress of mathematicians 2018, ICM 2018. Volume III. Invited lectures. Hackensack, NJ: World Scientific; Rio de Janeiro: Sociedade Brasileira de Matemática (SBM). 2363–2390 (2018; Zbl 1448.35398);
T. D. Drivas and T. M. Elgindi, EMS Surv. Math. Sci. 10, No. 1, 1–100 (2023; Zbl 1532.35359);
P. Constantin, Bull. Am. Math. Soc., New Ser. 44, No. 4, 603–621 (2007; Zbl 1132.76009)].
In
[Ann. Math. (2) 194, No. 3, 647–727 (2021; Zbl 1492.35199)]
(see also [T. M. Elgindi et al., “On the stability of self-similar blow-up for \(C^{1,\alpha}\) solutions to the incompressible Euler equations on \(\mathbb{R}^3\)”, Preprint, arXiv:1910.14071]),
T. 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
[J. Chen and T. Y. Hou, Commun. Math. Phys. 383, No. 3, 1559–1667 (2021; Zbl 1485.35071)]
in the same setting as the Hou-Luo blowup scenario
[G. Luo and T. Y. Hou, Proc. Natl. Acad. Sci. USA 111, No. 36, 12968–12973 (2014; Zbl 1431.35115);
Multiscale Model. Simul. 12, No. 4, 1722–1776 (2014; Zbl 1316.35235)].
On the other hand, the authors of
[A. F. Vasseur and M. Vishik, Commun. Math. Phys. 378, No. 1, 557–568 (2020; Zbl 1446.35114)] and
[L. Lafleche et al., J. Math. Pures Appl. (9) 155, 140–154 (2021; Zbl 1484.76016)]
recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in
[Vasseur and Vishik, loc. cit.] and
[Lafleche et al., loc. cit.]
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 [loc. cit.],
Chen and Hou [loc. cit.],
Vasseur and Vishik [loc. cit.] and
Lafleche et al. [loc. cit.] 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 [loc. cit.] and
Lafleche et al. [loc. cit.].
These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions.
MSC:
| 35Q31 | Euler equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B35 | Stability in context of PDEs |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76E30 | Nonlinear effects in hydrodynamic stability |
| 35A21 | Singularity in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Citations:
Zbl 0983.76001; Zbl 1143.76389; Zbl 1448.35398; Zbl 1532.35359; Zbl 1132.76009; Zbl 1492.35199; Zbl 1485.35071; Zbl 1431.35115; Zbl 1316.35235; Zbl 1446.35114; Zbl 1484.76016References:
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