Guo Luo

Potentially singular solutions of the 3D axisymmetric Euler equations

Significance Whether infinitely fast spinning vortices can develop in initially smooth, incompressible inviscid flow fields in finite time is one of the most challenging problems in fluid dynamics. Besides being a difficult mathematical question that has remained open for more than 250 years, the problem also attracts great attention in the physics and engineering communities due to its potential connection to the onset of turbulence in viscous flows. This paper attempts to provide an affirmative answer to this long-standing open question from a numerical point of view, by describing a class of rotationally symmetric flows from which infinitely fast spinning vortices can form in finite time. It suggests, after decades of controversies, a promising direction to the resolution of the problem. 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 τ2 = 0.003505 at which time the vorticity amplifies by more than (3 × 108)-fold and the maximum mesh resolution exceeds (3 × 1012)2. The vorticity vector is observed to maintain four significant digits throughout the computations.

LV shape and motion: B-spline-based deformable model and sequential motion decomposition

In this paper, we extend a previous work by J. Park and propose a uniform framework to reconstruct left ventricle (LV) geometry/motion from tagged MR images. In our work, the LV is modeled as a generalized prolate spheroid, and its motion is decomposed into four components-global translation, polar radial/z-axis compression, twisting, and bending. By formulating model parameters as tensor products of B-splines, we develop efficient algorithms to quickly reconstruct LV geometry/motion from extracted boundary contours and tracked planar tags. Experiments on both synthesized and in vivo data are also reported.

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.

Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier–Stokes (NS) system and show how the IE approach can give rise to local existence proofs. In this approach, the global existence problem in three-dimensional NS systems, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/t or some positive power of 1/t). Furthermore, the errors in numerical computations in the associated IE can be controlled rigorously, which is very important for nonlinear PDEs such as NS when solutions are not known to exist globally. Moreover, computation of the solution of the IE over an interval [0,p0] provides sharper control of its p→∞ behaviour. Preliminary numerical computations give encouraging results.

Integral formulation of 3D Navier-Stokes and longer time existence of smooth solutions

We consider the 3-D Navier-Stokes initial value problem,

On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations

v_t - \nu \Delta v = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in \mathbb{T}^3 (*)

Wiener Type Regularity of a Boundary Point for the 3D Lamé System

where

An integral equation approach to smooth 3D Navier–Stokes solution

\mathcal{P}