# Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

@article{Chen2021FiniteTB,
title={Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with \$\$C^\{1,\alpha \}\$\$ Velocity and Boundary},
author={Jiajie Chen and Thomas Y. Hou},
journal={Communications in Mathematical Physics},
year={2021},
volume={383},
pages={1559-1667}
}
• Published 1 October 2019
• Mathematics
• Communications in Mathematical Physics
Inspired by the recent numerical evidence of a potential 3D Euler singularity \cite{luo2013potentially-1,luo2013potentially-2}, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in \cite{luo2013potentially-1…
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## References

SHOWING 1-10 OF 41 REFERENCES

### Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

• Mathematics
Annals of PDE
• 2019
For all $\epsilon>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $\{(x,y,z)\in\mathbb{R}^3: (1+\epsilon|z|)^2\leq x^2+y^2\}$

### On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations

• Mathematics
Communications on Pure and Applied Mathematics
• 2019
We present a novel method of analysis and prove finite time asymptotically self‐similar blowup of the De Gregorio model [13, 14] for some smooth initial data on the real line with compact support. We

### On the Effects of Advection and Vortex Stretching

• Mathematics
Archive for Rational Mechanics and Analysis
• 2019
We prove finite-time singularity formation for De Gregorio’s model of the three-dimensional vorticity equation in the class of $$L^p\cap C^\alpha (\mathbb {R})$$ L p ∩ C α ( R ) vorticities for some

### Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System

• Mathematics
Annals of PDE
• 2020
As a follow up to our work [ 27 ], we give examples of finite-energy and Lipschitz continuous velocity field and density $$(u_0,\rho _0)$$ ( u 0 , ρ 0 ) which are $$C^\infty$$ C ∞ -smooth away from

### Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

• Mathematics
Multiscale Model. Simul.
• 2014
A local analysis near the point of the singularity suggests the existence of a self-similar blowup in the meridian plane and a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries.

### Singularity formation and global Well-posedness for the generalized Constantin–Lax–Majda equation with dissipation

We study a generalization due to De Gregorio and Wunsch et al of the Constantin–Lax–Majda equation (gCLM) on the real line where H is the Hilbert transform and . We use the method in Chen J et al

### On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations

• Mathematics
• 2013
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a

### Spatial Profiles in the Singular Solutions of the 3D Euler Equations and Simplified Models

The partial differential equations (PDE) governing the motions of incompressible ideal fluid in three dimensional (3D) space are among the most fundamental nonlinear PDEs in nature and have found a

### Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations

• T. Hou
• Mathematics
Acta Numerica
• 2009
Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite-time singularity from smooth initial data with finite energy has been one of the most long-standing open

### Potentially singular solutions of the 3D axisymmetric Euler equations

• Physics
Proceedings of the National Academy of Sciences
• 2014
This paper attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by describing a class of rotationally symmetric flows from which infinitely fast spinning vortices can form in finite time.