3D Axisymmetric Euler Equations Research Articles

Remarks on the smoothness of the C1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

We show that the constructions of C1,α asymptotically self-similar singularities for the three-dimensional (3D) Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity u∈C1,α that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted Hölder estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a C1,α singularity for the 3D axisymmetric Euler equations without swirl and with velocity u∈C∞(R3∖{0}) .

Axi-symmetric solutions for active vector models generalizing 3D Euler and electron–MHD equations

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂tB+V⋅∇B=B⋅∇V,V=−∇×(−Δ)−aB,∇⋅B=0,where B is a time-dependent vector field in R3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1/2<a<1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.

Potential Singularity of the 3D Euler Equations in the Interior Domain

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo and Hou (111:12968–12973, 2014) and (12:1722–1776, 2014), which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou and Huang in ( arXiv:2102.06663 , 2021) and (435:133257, 2022). One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in Hou and Huang ( arXiv:2102.06663 , 2021) and (435:133257, 2022). More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier–Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.

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

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.

On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl

We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $\omega^\theta/r$ plays the role of vorticity in 2D. First, we prove that the gradient of $\omega^\theta/r$ can grow at most double exponentially with improving a priori bound close to the axis of symmetry. Next, on the unit ball, we show that at the boundary, one can achieve double exponential growth of the gradient of $\omega^\theta/r$.

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

The growth of smooth solutions to 2D inviscid Boussinesq equations without heat conduction and the 3D axisymmetric Euler equations was investigated, to find regions where these systems have fast growing solutions. Through appropriately choosing the initial temperature and the velocity component, the Boussinesq system was decoupled into 2 parts. From the part involving only the vorticity, the vorticity and velocity can be solved and the smooth regions determined. From the part involving the temperature, one can see that the growth of temperature derivatives depends only on the velocity component. Through choosing that component appropriately, solutions with temperature derivatives of exponential growth were constructed on certain unbound smooth regions. The same method was applied to the axisymmetric Euler equations. Through choosing the radial velocity component appropriately, the system can be decoupled and one can ultimately find a class of smooth domains, and on them smooth global solutions of exponential growth. This investigation extends the results of Chae, Constantin and Wu on the inviscid Boussinesq system without heat conduction on a 2D cone to a class of smooth domains. Their method was also applied to the 3D axisymmetric Euler equations to obtain a similar result.

SIGEST

The SIGEST article in this issue is “Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study,” by Guo Luo and Thomas Y. Hou. This work considers the nonexistence of global finite-energy solutions to the 3D Euler equations. The authors propose a class of potentially singular solutions for the axisymmetric case, and present compelling numerical evidence that finite-time blowup occurs. Their contribution is closely related to the Clay Millennium Prize Problem on existence and smoothness of solutions to the Navier--Stokes equation, and hence sheds light on a preeminent open problem in fluid mechanics. At the heart of this work is a meticulous set of computational experiments that use customized Galerkin approximations and adaptive meshes in order to track a nearly singular numerical solution with high accuracy. The numerical results are validated with a range of tests, and the behavior close to the putative point of singularity is used to propose asymptotic scalings for various important quantities. The experiments also lead to a relevant 1D reduced model for which a finite-time singularity can be established rigorously. The original 2014 article, which appeared in SIAM's Multiscale Modeling and Simulation journal, has attracted considerable attention. In preparing this SIGEST version, the authors have made numerous modifications and revisions. These include: expanding the introductory section, streamlining and condensing the presentation of the numerical results, and updating the reference list. They have also highlighted various directions in which these equations, first published in 1757 by Leonhard Euler, may continue to tantalize both applied and numerical analysts.

Formation of Finite-Time Singularities in the 3D Axisymmetric Euler Equations: A Numerics Guided Study

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...

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

It is shown that the self-similar ansatz proposed by T. Hou and G. Luo to describe a singular solution of the 3D axisymmetric Euler equations leads, without assuming any asymptotic condition on the self-similar profiles, to an over-determined system of partial differential equations that produces two families of solutions: a class of trivial solutions in which the vorticity field is identically zero, and a family of solutions that blow-up immediately, where the vorticity field is governed by a stationary regime. In any case, the analytical properties of these solutions are not consistent with the numerical observations reported by T. Hou and G. Luo. Therefore, this result is a refinement of the previous work published by D. Chae and T.-P. Tsai on this matter, where the authors found the trivial class of solutions under a certain decay condition of the blow-up profiles.

Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations

We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which approximates the dynamics of the Euler equations on the solid boundary of a cylindrical domain. We prove the existence of a discrete family of self-similar profiles for this model and analyze their far-field properties. The self-similar profiles we find are consistent with direct simulation of the model and enjoy some stability property.

Potentially singular solutions of the 3D axisymmetric Euler equations.

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 ∼(ts - t)(-2.46), where ts ∼ 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.

Viscous approximation and weak solutions of the 3D axisymmetric Euler equations

Keywords: axisymmetric 3D Euler equations; global existence; viscous approximation; weak solutions

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

We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.

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

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 x 10^(12))^2 near the point of the singularity, we are able to advance the solution up to tau_2 = 0.003505 and predict a singularity time of t(s) approximate to 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector. and observing a (3 x 10^8)-fold increase in the maximum vorticity parallel to omega parallel to(infinity). 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.

Statistical mechanics of the 3D axisymmetric Euler equations in a Taylor–Couette geometry

In the present paper, microcanonical measures for the dynamics of three dimensional (3D) axially symmetric turbulent flows with swirl in a Taylor–Couette geometry are defined, using an analogy with a long range lattice model. We compute the relevant physical quantities and argue that two kinds of equilibrium regime exist, depending on the value of the total kinetic energy. For low energies, the equilibrium flow consists of a purely swirling flow whose toroidal profile depends on the radial coordinate only. For high energies, the typical toroidal field is uniform, while the typical poloidal field is organized into either a single vertical jet or a large scale dipole, and exhibits infinite fluctuations. This unusual phase diagram comes from the poloidal fluctuations not being bounded for the axisymmetric Euler dynamics, even though the latter conserve infinitely many ‘Casimir invariants’. This shows that 3D axially symmetric flows can be considered as intermediate between 2D and 3D flows.

On the Lagrangian dynamics of the axisymmetric 3D Euler equations

We study the dynamics along the particle trajectories for the 3D axisymmetric Euler equations. In particular, by rewriting the system of equations we find that there exists a complex Riccati type of structure in the system on the whole of R 3 , which generalizes substantially the previous results in [5] (D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity 21 (2008) 2053–2060). Using this structure of equations, we deduce the new blow-up criterion that the radial increment of pressure is not consistent with the global regularity of classical solution. We also derive a much more refined version of the Lagrangian dynamics than that of [6] (D. Chae, On the Lagrangian dynamics for the 3D incompressible Euler equations, Comm. Math. Phys. 269 (2) (2007) 557–569) in the case of axisymmetry.

On the global well-posedness for the axisymmetric Euler equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces \({B_{p,1}^{1+3/p}}\). In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity.

Smooth Approximations and Exact Solutions of the 3D Steady Axisymmetric Euler Equations

In this paper, we prove that a class of C1-smooth approximate solutions {ue, pe} to the 3D steady axisymmetric Euler equations will converge strongly to 0 in \({L^2_{loc}(R^3)}\) . The main assumptions are that the approximate solutions have uniformly finite energy and approach a constant state at far fields. We also show a Liouville type theorem that there are no non-trivial C1-smooth exact solutions with finite energy and uniform constant state at far fields.

On the global existence for the axisymmetric Euler equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in some critical Besov spaces.

Singular solutions to the 3D axisymmetric incompressible Euler equations

The problem of development of singular solutions of the 3D Euler equations is considered in the particular case of flows with an axis of symmetry. There is a strong analogy between the physics of such flows, and Boussinesq convection in two dimensions, the buoyancy being replaced by the centripetal acceleration. A hot bubble, initially at rest in cold fluid, tends to rise. During the evolution, strong gradients of temperature develop near the cap of the bubble, while on the sides, vortex sheets tend to roll up. When the cap of the bubble starts folding, a rapid growth of the gradients is observed, suggesting a singularity of the vorticity in the 3D axisymmetric flows like |ω| max ≈ 1/( t ∗ − t) 2. Analytic estimates for the rate of stretching, consistent with our numerical observations, are provided.