Abstract: 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 viscosity develop a finite time singularity.

Bio: Thomas Yizhao Hou is the Charles Lee Powell professor of applied and computational mathematics at Caltech. His research interests include 3D Euler singularity, interfacial flows, multiscale problems, and data science. He received his Ph.D. from UCLA in 1987, and joined the Courant Institute as a postdoc in 1987. He became a tenure track assistant professor at the Courant Institute in 1989 and then was promoted to tenured associate professor in 1992. He moved to Caltech in 1993, served as the department chair of applied and computational mathematics from 2000 to 2006, and was named the Charles Lee Powell Professor in 2004. Dr. Hou has received a number of honors and awards, including Fellow of American Academy of Arts and Sciences in 2011, a member of the inaugural class of SIAM Fellows in 2009 and AMS Fellows in 2012, the Computational and Applied Sciences Award from USACM in 2005, the Morningside Gold Medal in Applied Mathematics in 2004, the SIAM Wilkinson Prize in Numerical Analysis and Scientific Computing in 2001, the Frenkiel Award from the Division of Fluid Mechanics of American Physical Society in 1998, the Feng Kang Prize in Scientific Computing in 1997, a Sloan fellow from 1990 to 1992. He was also the founding Editor-in-Chief of the SIAM Journal on Multiscale Modeling and Simulation from 2002 to 2007.

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Link: https://zoom.com.cn/j/86860286785?pwd=QlIySEZ0aUtQQTVuQmxoTld4ZlJHZz09