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

Abstract

Abstract. Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 × 10 12 ) 2 near the point of the singularity, we are able to advance the solution up to τ 2 = 0.003505 and predict a singularity time of ts ≈ 0.0035056, while achieving a pointwise relative error of O(10 −4 ) in the vorticity vector ω and observing a (3 × 10 8 )-fold increase in the maximum vorticity ω ∞ . The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

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

“…First we describe how the model originally came about and then we exhibit the specific solutions to the model which we will be using later on. We remark that this model can also be used to model situations other than the one discussed in Section 2; in fact, I believe that some form of this model is also behind the singularity in the numerical work [44].…”

“…There are numerous previous works on the global regularity problem and we will only discuss a few which are directly relevant to this work. A more extensive list of works can be found in the book [45], the review papers [26], [2], [8], and [36], the numerical work [44] as well as the author's work with I. Jeong [24]. We will discuss three types of results here: blow-up criteria, infinite-time singularity formation, and model problems.…”

“…One work in this direction which we are drawing inspiration from is [22] where it is shown that the regularizing effect of the advection term can be minimized by considering vorticity at C α regularity with α small. After the numerical work of Luo and Hou [44] and the work of Kiselev and Šverák [38], several other model problems related to the scenario in [44] were considered (see [7], [37], for example). One of the ideas in these works is to study scenarios where the Biot-Savart law (1.6) can be decomposed into a main singular term and a more regular lower order term.…”

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

“…Based on the numerical results about potential singularity profile for 3D axisymmetric Euler equation ( [12]), we are particularly interested in the case when ω is periodic in x (formerly z) variable and will treat this case in the next section. The periodic assumption is not crucial; in the appendix we will outline the arguments which adjust the proof to the real line case.…”

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

“…In this paper we study a one-dimensional model introduced in [20] (also see [21]) in connection with incompressible Euler equations. The study of one-dimensional models for hydrodynamical equations has a long history, going back to the works of Burgers [4] and Hopf [17].…”