This project is aimed at investigating regularity and possible blow-up of solutions to partial differential equations of fluid mechanics and related mathematical models. Understanding of properties of these models is a fundamental challenge both for mathematics and science. The regularity and stability of the solutions are strongly related to complex multiscale physical phenomena such as turbulence and phase transitions. A principal goal of this project is developing new techniques to tackle the problem of global regularity for super-critical systems including reaction-diffusion systems or the 3D Navier-Stokes system.
This research program will make some significant progress in the fundamental understanding of the mathematics involved in fluid mechanics and chemistry. It will help also to train new graduate students in this important field. The discovery of mathematical structure in physical models will lead to a deeper understanding of the physical phenomena themselves. It will also provide some information about the limits of validity of the theoretical physical models that are used to describe physical phenomena.
Major outcomes: We summarize, in the following, the major findings obtained in the three main areas of the project. 1) Study of non local operators: A most acclaimed result on regularity of solutions to transport equation with non local effects was obtained. Especially, a conjecture about the global regularity of solutions to the surface quasi-geostrophic equation in the critical case has been solved. It has been identified by Thomson Reuters Essential Science IndicatorsSM as a featured New Hot Paper in the field of Mathematics, which means it is one of the most-cited papers in this discipline published during the past two years. It has already be cited more than 100 times. 2) Regularity theory for fluid mechanics: A robust blow-up technique for the Navier-Stokes equation has been developed to study quantitative properties of the Navier-Stokes equations. It allows, down the road, to obtain new estimates on higher derivatives of weak solutions. 3) Strong stability of shocks for conservation laws: This project enjoys some surprising breakthrough. The aim was to obtain a L^2 theory (more refined than the standard L^1 theory) to study the structure of the discontinuities (known as shocks) in inviscid compressible fluid mechanics (as the compressible Euler equation), which is part of a family of equations called conservation laws. The PI and his collaborators obtained a surprising result of strong stability (up to translation) in L^2 for such shocks. Training and Development: 4 graduate student worked on this project. One advanced student works in tandem with the PI in advance theory for conservation laws. The three others, worked in the development of recent tools, one on compressible Navier-Stokes equation, and two on incompressible Navier-Stokes equation. 3 Post-docs have been actively included in this program (even if they were not supported by the project nor officially advised by the PI). One is a former student of the PI, now Assistant Professor in Taiwan. He developed his skills on regularity theory for fractional Laplacian.
