Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with C^{1 ,α} Velocity and Boundary
Abstract
Inspired by the numerical evidence of a potential 3D Euler singularity by LuoHou [30, 31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with C^{1 ,α} initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with C^{1 ,α} 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 [30, 31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use C^{1 ,α} initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate selfsimilar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with C^{1 ,α} initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 May 2021
 DOI:
 10.1007/s00220021040671
 arXiv:
 arXiv:1910.00173
 Bibcode:
 2021CMaPh.383.1559C
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 There is an oversight about the $C^{1,\alpha}$ regularity of the initial velocity for the 3D Euler equations in the previous version of this paper. This oversight has been fixed easily with minor changes in the construction of the approximate steady state and the truncation of the approximate steady state. These changes do not affect the nonlinear stability estimates of the 3D Euler equations