Thomas Yizhao Hou, California Institute of Technology
Potential singularity of
3D incompressible Euler equations and the nearly singular behavior of 3D Navier-Stokes
equations
Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In an effort to provide a rigorous proof of the potential Euler singularity revealed by Luo-Hou's computation, we develop a novel method of analysis and prove that the original De Gregorio model and the Hou-Lou model develop a finite time singularity from smooth initial data. Using this framework and some techniques from Elgindi's recent work on the Euler singularity, we prove the finite time blowup of the 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ initial velocity and boundary. Further, we present some new numerical evidence that the 3D incompressible Euler equations with smooth initial data develop a potential finite time singularity at the origin, which is quite different from the Luo-Hou scenario. Our study also shows that the 3D Navier-Stokes equations develop nearly singular solutions with maximum vorticity increasing by a factor of $10^7$. However, the viscous effect eventually dominates vortex stretching and the 3D Navier-Stokes equations narrowly escape finite time blowup. Finally, we present strong numerical evidence that the 3D Navier-Stokes equations with slowly decaying time-dependent viscosity develop a finite time singularity.
Mei-Chi Shaw, University of Notre Dame
The Cauchy-Riemann Equations in Complex
Analysis
Holomorphic functions are solutions to the homogeneous Cauchy-Riemann equations. The profound influence of partial differential equations on complex analysis began from Riemann's proof of the Mapping Theorem with the Dirichlet problem. In this talk, we discuss the important role that the inhomogeneous Cauchy-Riemann equations play in one and several complex variables. The emphasis is on the recent progress on domains in complex manifolds.
Chia-Fu Yu, Institute of Mathematics, Academia Sinica
Mass formula and Oort's
conjecture for supersingular abelian threefolds
The problem we study in this talk is originated from works of Eichler and Deuring
who proved a beautiful correspondence between supersingular elliptic curves and ideal classes of a
definite quaternion algebra over the rational numbers.
Its generalization to supersingular abelian varieties has been studied by Ekedahlm Ibukiyama,
Katsura, K.-Z. Li, Oort and others.
We compute the mass of each orbit of supersingular abelian threefolds with same crystalline
cohomology. This decomposes the supersingular locus into strata which play a key role for jumps of
automorphism groups and yields a solution to a conjecture of Oort. This is based on joint work with
Valentijn Karemaker and Fuetaro Yobuko.