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

Abstract

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,\alpha}$ 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 [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 $C^{1,\alpha}$ initial velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed 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 self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^{\theta})^2, u_0^r, u_0^z \in C^{1,\alpha}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,\alpha}$, our interpretation that the velocity is $C^{1,\alpha}$ is not correct since the velocity in 3D also depends on $u^{\theta}$, which is not $C^{1,\alpha}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.