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 30] and the recent breakthrough by Elgindi [10] 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 [29,30] 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 [10]. We also use some strategy proposed in our recent joint work with Huang in [6] and adopt several methods of analysis in [10] 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 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.

On the regularity of the De Gregorio model for the 3D Euler equations

“…In particular, in [5], we established that the gCLM model on S 1 with a slightly less than 1, which can be seen as a slight perturbation to (1.2), develops finite time singularity from some smooth initial data in X. Thirdly, this scenario can be seen as a 1D analog of the hyperbolic blowup scenario for the 3D Euler equations reported by Hou-Luo [32,33]. See also [7,28,29]. In fact, the restriction of the (angular) vorticity in [7,28,29,32,33] to the boundary has the same sign and symmetry properties as those in X.…”

“…Inspired by the anisotropic property of θ in [7], i.e. |θ y | << |θ x | near the origin, we drop the θ y term.…”

“…In both cases, θ x vanishes linearly in y, and thus the effect of y−advection can be an obstacle for singularity formation. Such effect can be overcome by imposing a solid boundary on y = 0 and singularity formation with C 1,α velocity has been established in [7]. For smooth data, the importance of boundary has been studied in [32,33].…”

“…See also [7,28,29]. In fact, the restriction of the (angular) vorticity in [7,28,29,32,33] to the boundary has the same sign and symmetry properties as those in X. Thus, to establish global regularity of (1.2) for general smooth initial data, we need to address the important question of whether there is a finite time blowup in this class.…”

On Magnetic Inhibition Theory in Non-resistive Magnetohydrodynamic Fluids

“…We remark that such global stability result is not excepted for the 3D incompressible Euler equations, i.e., the magnetic field is absent in the system (1.2). Interesting readers can refer to [9,35] and [30,50] for the singularity formation of solutions and the solutions with double-exponential growth in Euler equations, resp.. Bardos-Sulem-Sulem's large perturbation result can be roughly described as follows: We naturally make an association with the following well-known assertion in viscous (pure) fluids (see [12,17,33] for examples): The above two assertions present that magnetic fields can inhibit the singularity formation of solutions with large initial velocity as well as viscosity, though the physical mechanisms of inhibition/stabilzing are different. Based on the above two assertions, we easily further believe that such large perturbation result shall also exists for the hyperbolic-parabolic system (1.1), that is (roughly speaking):…”