Whether the 3D 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 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 tau_2 = 0.003505 and predict a singularity time of t_s ~ 0.0035056, while achieving a pointwise relative error of O(10^(-4)) in the vorticity vector and observing a (3*10^8)-fold increase in the maximum vorticity. The numerical data is 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. We also discuss a 1D model which can be viewed as a local approximation to the Euler equations near the solid boundary of the cylinder. The finite-time blowup of this 1D model is proved for a class of smooth initial data, which are essentially restrictions of the initial data used in the full 3D blowup calculations.

罗果博士为中山大学学士，香港中文大学硕士，美国俄亥俄州立大学(Ohio State University)博士。毕业后在加州理工大学从事博士后研究，现在为香港城市大学数学系Assistant Professor。他的主要研究方向为偏微分方程的计算及理论。