Potentially singular solutions of the 3D axisymmetric Euler equations

Abstract

The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t s − t) −2.46 , where t s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ 2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10 8 )-fold and the maximum mesh resolution exceeds (3 × 10 12 ) 2 . The vorticity vector is observed to maintain four significant digits throughout the computations.W hether initially smooth solutions to the 3D incompressible Euler equationscan develop a singularity in finite time is one of the most fundamental problems in mathematical fluid dynamics. Standing open for more than 250 y and closely related to the Clay Millennium Prize Problem on the Navier-Stokes equations, the problem has received great attention from not only the mathematics but also the physics and engineering communities, where the formation of singularities in inviscid (Euler) flows is believed to be relevant to the creation of small scales in viscous turbulent flows (1-3). The finite-time blowup problem has been studied extensively from both mathematical and numerical points of view. On the mathematical side, a number of useful blowup/nonblowup criteria have been obtained over the years, which have greatly facilitated the numerical search of a finite-time singularity (4-10). On the numerical side, interesting numerical simulations suggesting the existence of a finite-time singularity have been reported from time to time (see, for example, refs. 11-17), but in essentially every case, evidence against a singularity was found in subsequent studies (see, for example, refs. 18-21). This casts doubt on the validity of the claimed singularity and leaves the question of finite-time blowup an unresolved puzzle. By focusing on flows with rotational symmetry and other special properties, we have identified, through careful numerical studies, a class of potentially singular solutions to the 3D axisymmetric Euler equations in a radially bounded, axially periodic cylinder (see Eqs. 2 and 3 below). The simulations take advantage of the reduced computational complexity in cylindrical geometries, and use a specially designed adaptive mesh to achieve a maximum mesh resolution of over (3 × 10 12 ) 2 near the point of the singularity. This results in a computed vorticity vector with four digits of accuracy (up to the stopping time) and with a (3 × 10 8 )-fold increase in magnitude. The numerical data are checked against all major blowup/nonblowup criteria to confirm the validity of the singularity. A careful local analysis also suggests the existence of a self-similar blowup in the meridian plane.We...

An Incompressible 2D Didactic Model with Singularity and Explicit Solutions of the 2D Boussinesq Equations

“…Very recently Luo and Hou performed careful numerical simulations of the 3D axisymmetric incompressible Euler equations which suggested the appearance of a finite-time singularity [9]. We briefly describe the setup and their main result.…”

“…The velocity on the unit circle z = 0, r = 1 vanishes, and thus all points on this circle are stagnation points. According to [9], the vorticity of a numerical solution at the stagnation points blows up in finite time. This finite-time singularity does not appear to be well understood theoretically.…”

Remark on Luo-Hou’s Ansatz for a Self-similar Solution to the 3D Euler Equations

“…Since the self-similar ansatz of Luo-Hou [1] is numerically observed, it is robust in some sense. One possible way to explain the discrepancy between [1] and our results is that the self-similar singularity is only observed in [1] in a subregion of a timedependent window W δ(t) defined in (2.6), with…”

“…For a possible blow-up scenario at the circle on the boundary of the cylinder, observed numerically in [1], Luo-Hou [1, §4.7] proposed the following self-similar ansatz for the solutions to (1.5)-(1.7),…”

“…The region D ∞ (t) of self-similarity studied in [1] is defined dynamically as where the vorticity magnitude exceeds one half of its maximal magnitude at each time t. It is observed numerically to shrink to the boundary circle as t → T − . If (2.1)-(2.3) are valid in the set D ∞ (t), the diameter of D ∞ (t) should be proportional to (T − t) γ and corresponds to a fixed finite set in the left half RZ-plane.…”