Date:December 16 - 18

Venue:E4-233 & E10-212, Yungu Campus

ZOOM ID:964 7160 6669

Passcode:658979

Ⅰ Organizers

Alexey Cheskidov，Mimi Dai

Ⅱ Agenda

Date	Time	Venue	Title	Speaker
12.16	9:00-10:00	E10-212	Pathological solutions for active scalar equations	Mimi Dai
	10:00-11:00		A class of singular SPDEs via convex integration	Xiangchan Zhu
	11:00-12:00		Non-unique ergodicity for 3D Navier-Stokes and Euler equations	Rongchan Zhu
	12:00-13:00		Lunch
	13:00-13:20		Choosing Open Problems
	13:20-17:00		Discussing and Solving Problems
	17:30		Dinner
12.17	9:00-10:00	E10-212	Existence and smoothness of solution of the Navier-Stokes equation	Huashu Dou
	10:00-11:00		Structure of helicity and a class of global smooth solutions to 3D imcopressible Navier-Stokes equations	Yi Zhou
	11:00-12:00		Strongly compact strong trajectory attractors for the 3D Navier-Stokes equations	Songsong Lu
	12:00-13:00		Lunch
	13:00-13:20		Choosing Open Problems
	13:20-17:00		Discussing and Solving Problems
	17:30		Dinner
12.18	9:00-10:00	E4-233	Quantitative partial regularity of the Navier-Stokes equations and applications	Xiao Ren
	10:00-11:00		TBA	Alexey Cheskidov
	11:00-12:00		Stability of shear flows in viscous fluids	Weiren Zhao
	12:00-13:00		Lunch
	13:00-13:20		Choosing Open Problems
	13:20-17:00		Discussing and Solving Problems
	17:30		Dinner

Ⅲ Talks

1. 9:00-10:00, December 16, Saturday

Speaker:Mimi Dai, University of Illinois at Chicago

Title:Pathological solutions for active scalar equations

Abstract:We will discuss recent development in construction of weak solutions for active scalar equations, for which the uniqueness and certain conservation law are violated. The purpose is to verify the sharp regularity threshold that separates the rigidity and flexibility regimes.

2. 10:00-11:00, December 16, Saturday

Speaker:Xiangchan Zhu, Chinese Academy of Sciences

Title:A class of singular SPDEs via convex integration

Abstract:In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method.

In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most $-1/2-\kappa$ for any $\kappa>0$. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in $L^{2}\cup B^{-1+\kappa}_{\infty,\infty}$, $\kappa>0$, and implies also non-uniqueness in law.

Finally I will show the existence, non-uniqueness, non-Guassianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.

3. 11:00-12:00, December 16, Saturday

Speaker:Rongchan Zhu, Beijing Institute of Technology

Title:Non-unique ergodicity for 3D Navier-Stokes and Euler equations

Abstract:We establish existence of infinitely many stationary solutions as well as ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. Moreover, we are able to make conclusions regarding the vanishing viscosity limit. The result is based on a new stochastic version of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.

4. 9:00-10:00, December 17, Sunday

Speaker:Huashu Dou, Zhejiang Sci-Tech University

Title:Existence and smoothness of solution of the Navier-Stokes equation

Abstract:Existence and smoothness of solution of the Navier-Stokes equation are exactly disproved for the first time by using two different approaches: Energy gradient theory and Poisson equation method. At a higher Reynolds number, the velocity profile in laminar flow is distorted under a disturbance and the velocity deficit is produced. It is found that the viscous term is zero instantaneously and velocity discontinuity occurs at a distorted position, which forms the singularity of Navier-Stokes equation. In addition, the singularity of the Navier-Stokes equation at the zero source term location is also confirmed by the analysis of the Poisson equation. The analytical results show that the singularity of the Navier-Stokes equation is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence, which is in agreement with experiments and simulations. Since the velocity is not differentiable at the singularity, there exist no smooth solutions of the Navier-Stokes equation in global domian at high Reynolds number (beyond laminar flow).

5. 10:00-11:00,December 17, Sunday

Speaker:Yi Zhou, Fudan University

Title:Structure of helicity and a class of global smooth solutions to 3D imcopressible Navier-Stokes equations

Abstract:I will report works by my collabotators and myself on the structire of helicity of 3D imcompressible Navier-Stokes equations and a class of large smooth solutions related to this on various spaces.

6. 11:00-12:00,December 17, Sunday

Speaker:Songsong Lu, Sun Yat-sen University

Title:Strongly compact strong trajectory attractors for the 3D Navier-Stokes equations

Abstract:We introduce a notion of a strongly compact strong trajectory attractor for the 3D Navier-Stokes equations, whose existence implies two properties. One is that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete bounded solutions on the global attractor are capable of uniformly approximating all Leray-Hopf weak solutions within the accuracy in the natural strong metric after sufficiently large time. The other one is the strong equicontinuity of all the complete bounded solutions on the global attractor. We prove the existence of such an attractor for the system is with a fixed normal force and every complete bounded solution is strongly continuous. The notion of a (weak) trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one. We developed a framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system nearly ten years ago. Very recently, the theory of trajectory attractors is further developed in the natural strong metric. The framework is general and can also be applied to other dissipative partial differential equations for which the uniqueness of solutions might not hold. We are expecting ardently more insight coming in to push forward.

7. 9:00-10:00, December 18, Monday

Speaker:Ren Xiao, Peking University

Title:Quantitative partial regularity of the Navier-Stokes equations and applications

Abstract:The classical Caffarelli-Kohn-Nirenberg theorem states that the 1d parabolic Hausdorff measure of the singular set of a suitable weak solution must vanish. Its proof relies on the absolute continuity of the dissipation energy, which is a non-quantitative fact. We develop a quantitative argument using the pigeonhole principle and improve the Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor. This further improves a result of Choe and Lewis (2000). Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending only on the natural energies of the solution. A number of applications will be discussed, including a new regularity criterion concerning the direction of vorticity. Based on joint work with Prof. Zhen Lei and Prof. Gang Tian.

8. 10:00-11:00, December 18, Monday

Speaker:Alexey Cheskidov, Westlake University

Title:TBA

Abstract:TBA

9. 11:00-12:00, December 18, Monday

Speaker:Weiren Zhao, NYU Abu Dhabi

Title:Stability of shear flows in viscous fluids

Abstract:In this talk, I will introduce some recent results on the asymptotic stability of shear flows in viscous fluids. First, I will discuss the two important stability mechanisms: inviscid damping and enhanced dissipation which can be obtained easily for Couette flow at linear level. Then, we focus on the nonlinear system and introduce the stability threshold problem for Couette flow. We will see how the regularity and size of perturbations will affect the stability of the system. Finally, I will talk about the stability problem for general monotone shear flows where the wave operator method is developed.