Date:June 4 - 8

Venue:E5-329, Yungu Campus

ZOOM ID:952 2949 3423

Passcode:004072

ⅠAgenda

Date	Time	Venue	Title	Speaker
6.5	10:00-11:00	E5-329	Rigidity of steady incompressible Euler flows with stagnation points and its applications	Chunjing Xie
	11:00-12:00		Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations	Yachun Li
	12:00-13:00		Lunch
	13:30-14:30		Potential multi-scale finite-time blowups of the incompressible Euler equations	De Huang
	14:30-15:30		Transition threshold problem for 3-D Couette flow in a finite channel	Zhifei Zhang
	15:30-16:30		Mixing flow and advection-diffusion-reaction equations	Xiaoqian Xu
	16:30-17:30		Singularity formation for some fluid equations	Mimi Dai
	17:30		Dinner
6.6	10:00-11:00	E5-329	Sharp non-uniqueness of weak solutions to some viscous fluid models	Zirong Zeng
	11:00-12:00		Global regularity for critical SQG in bounded domains	Quoc Hung Nguyễn
	12:00-13:00		Lunch
6.7	10:00-11:00	E5-329	Global solutions to the Euler-Coriolis system	Xiao Ren
	11:00-12:00		Illposedness for 3D fluids in supercritical spaces	Xiaoyutao Luo
	12:00-13:00		Lunch
	13:30-14:30		Remarks about supercritical SQG	Hongjie Dong
	14:30-17:00		Discussion
	17:30		Dinner
6.8	9:00-9:20	E5-329	Convergence rates in zero-relaxation limits for Euler-Maxwell systems near non-constant equilibrium	Rui Jin
	9:20-9:40		Self-similar finite-time blowups and traveling wave solutions of the generalized Constantin--Lax--Majda model for the 3D Euler equations	Xiang Qin
	9:40-10:00		Self-similar finite-time blowups of the Hou-Luo model	Xiuyuan Wang
	10:00-11:00		Uniform Bound of the Highest-order Energy of the 2D Incompressible Elastodynamics	Yuan Cai

Ⅱ Talks

1.10:00-11:00,June 5, Wednesday

Speaker:Chunjing Xie, Shanghai Jiao Tong University

Title:Rigidity of steady incompressible Euler flows with stagnation points and its applications

Abstract:Stagnation of flows is an interesting phenomenon in fluid mechanics. It induces many challenging problems in analysis. We first derive a Liouville type theorem for Poiseuille flows in the class of incompressible steady inviscid flows in an infinitely long strip, where the flows can have stagnation points. With the aid of this Liouville type theorem, we show the uniqueness of solutions with positive horizontal velocity for steady Euler system in a general nozzle when the flows tend to the horizontal velocity of Poiseuille flows at the upstream. Furthermore, this kind of flows are proved to exist in a large class of nozzles and we also prove the optimal regularity of boundary for the set of stagnation points. Finally, we give a classification of incompressible Euler flows via the set of flow angles.

2. 11:00-12:00,June 5, Wednesday

Speaker:Yachun Li, Shanghai Jiao Tong University

Title:Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes equations

Abstract:This talk concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. These results in particular yield that the Lions exponent is the sharp viscosity threshold for the uniqueness/non-uniqueness in law of Leray solutions. This is a joint work with Elia Brué, Rui Jin, and Deng Zhang.

3. 13:30-14:30,June 5, Wednesday

Speaker:De Huang, Peking University

Title:Potential multi-scale finite-time blowups of the incompressible Euler equations

Abstract:It remains an open problem whether the 3D incompressible Euler equations can develop finite-time singularity from smooth initial data in the whole space. Recent numerical results indicate the potential existence of self-similar finite-time blowups with multi-scale features. Different from the conventional one-scale blowup that has been extensively studied for many models of the 3D Euler equations, this new type of blowup is closely related to traveling wave solutions and may provide a new approach to studying Euler singularity. We will first present some related numerical findings, and then we will rigorously show that multi-scale self-similar blowups can happen for a simple model of the 3D Euler equations.

4.14:30-15:30,June 5, Wednesday

Speaker:Zhifei Zhang, Peking University

Title:Transition threshold problem for 3-D Couette flow in a finite channel

Abstract:The transition threshold problem was proposed by Trefethen et al. (Science 1993) with a goal toward understanding the transition mechanism of laminar flows. This talk is to deliver a stability threshold result of the Couette flow for the 3-D Navier-Stokes equations in a finite channel with non-slip boundary condition. This result should be optimal according to the asymptotic analysis result by Chapman.

5.15:30-16:30,June 5, Wednesday

Speaker:Xiaoqian Xu, Duke Kunshan University

Title:Mixing flow and advection-diffusion-reaction equations

Abstract:In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of applications is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation, such as the famous Keller-Segel equation for chemotaxis. I will also discuss about the examples of such flows in this talk.

6.16:30-17:30,June 5, Wednesday

Speaker:Mimi Dai, University of Illinois at Chicago

Title:Singularity formation for some fluid equations

Abstract:Finite time singularity formation for fluid equations will be discussed. Built on extensive study of approximate models, breakthroughs on this topic have emerged recently for the Euler equation and the surface quasi-geostrophic equation. Inspired by these works, we attempt to understand this challenging issue for the magnetohydrodynamics (MHD). Finite time singularity scenarios are discovered for the MHD in some contexts. The investigation also reveals connections between the MHD, Euler equation, and the surface quasi-geostrophic equation.

7.10:00-11:00, June 6, Thursday

Speaker:Zirong Zeng, Nanjing University of Aeronautics and Astronautics

Title:Sharp non-uniqueness of weak solutions to some viscous fluid models

Abstract:In this talk, I will present our recent results about non-uniqueness of weak solutions to some viscous fluid models. Concerning the incompressible hyper viscous Navier-Stokes equations, we proved the sharp non-uniqueness of weak solutions at two endpoints of the Ladyžhenskaya-Prodi-Serrin (LPS) criteria even in the hyper-viscous regime. For incompressible MHD equations, we prove the sharp non-uniqueness near one endpoint of the LPS condition. Furthermore, the strong vanishing viscosity and resistivity result is obtained, which yields the failure of Taylor’s conjecture along some sequence of weak solutions. Our proof is based on the spatial-temporal intermittent convex integration scheme. These are joint works with Yachun Li, Peng Qu and Deng Zhang.

8.11:00-12:00, June 6, Thursday

Speaker:Quoc Hung Nguyễn, Chinese Academy of Sciences

Title:Global regularity for critical SQG in bounded domains

Abstract:The critical surface quasi-geostrophic equations (SQG) are atmoshperic science equations which are widely studied as concise two dimensional models of rapid formation of small scales in incompressible fluids. When considered in the whole space or on the torus, the equations have been proved to have global smooth solutions more than fifteen years ago by Caffarelli-Vasseur and, independently, by Kiselev-Nazarov-Volberg. The problem of existence and uniqueness of global smooth solutions in bounded domains remained open until now. After giving some background, I will present a proof of global regularity obtained recently with Peter Constantin and Mihaela Ignatova.

9.10:00-11:00, June 7, Friday

Speaker:Xiao Ren, Peking University

Title:Global solutions to the Euler-Coriolis system

Abstract:We prove the global well-posedness and scattering for the 3D incompressible Euler-Coriolis system with sufficiently small, regular and localized initial data. Equivalently, we obtain global asymptotic stability for the uniform rotating solutions to the pure Euler system. This extends a recent work of Guo, Pausader and Widmayer to the general non-axisymmetric setting. Joint work with Prof. Gang Tian.

10.11:00-12:00, June 7, Friday

Speaker:Xiaoyutao Luo, Chinese Academy of Sciences

Title:Illposedness for 3D fluids in supercritical spaces

Abstract:I will show a construction of a class of smooth initial data that leads to instant norm inflation in supercritical Sobolev spaces for both the 3D Euler and Navier-Stokes equations.

11.13:30-14:30, June 7, Friday

Speaker:Hongjie Dong, Brown University

Title:Remarks about supercritical SQG

Abstract:The global wellposedness of the supercritical dissipative surface quasigeostrophic equation is an outstanding open problem. In this talk, I will present a couple of remarks about the equation. The first is a nonlinear instability result, while the second one is a stability result near radial profiles.

12.9:00-9:20, June 8, Saturday

Speaker:Rui Jin, Shanghai Jiao Tong University

Title:Convergence rates in zero-relaxation limits for Euler-Maxwell systems near non-constant equilibrium

Abstract:This talk concerns the simplification of the Euler-Maxwell system through the zero-relaxation limit towards drift-diffusion equations with non-constant doping functions. We carry out the global-in-time convergence analysis by establishing uniform estimates of solutions near non-constant equilibrium regarding the relaxation parameter and passing to the limit by using classical compactness arguments. Furthermore, we generalize stream function method to the non-constant equilibrium case, and together with the anti-symmetric structure of the error system and an induction argument, we establish global-in-time error estimates between smooth solutions to the Euler-Maxwell system and those to the drift-diffusion system, which are bounded by some power of the relaxation parameter.

13.9:20-9:40, June 8, Saturday

Speaker:Xiang Qin, Peking University

Title:Self-similar finite-time blowups and traveling wave solutions of the generalized Constantin--Lax--Majda model for the 3D Euler equations

Abstract:The fundamental question on the global regularity of the 3D Euler equations with smooth initial data of finite energy remains one of the most challenging open problems in fluid dynamics. The generalized Constantin–Lax–Majda model is a 1D model for the vorticity formulation of the 3D incompressible Euler equations, proposed to study the competitive relation between advection and vortex stretching. Recently, we have introduced an analytical method based on fixed-point theory, which is effective in establishing the existence and smoothness of self-similar blowups and traveling wave solutions across a wide range of the parameter. Our work unifies previous results and provides a new approach to studying potential singularity formation in the Euler equations and other partial differential equations.

14.9:40-10:00, June 8, Saturday

Speaker:Xiuyuan Wang, Shanghai Jiao Tong University

Title:Self-similar finite-time blowups of the Hou-Luo model

Abstract:The global regularity of the 3D Euler equations with smooth initial data remains one of the most challenging open problems in fluid dynamics. In 2014, the 1D Hou-Luo model was proposed to gain insight into the numerically observed self-similar singularity formation of the 3D axisymmetric Euler equations in a specific scenario. Our work introduces a purely analytic fixed-point method to prove that the Hou-Luo model admits exact self-similar finite-time blowup solutions with smooth profiles. Additionally, we provide evidence strongly suggesting that the Hou-Luo model may develop finite-time singularities with singular profiles.

15.10:00-11:00,June 8, Saturday

Speaker:Yuan Cai, Fudan University

Title:Uniform Bound of the Highest-order Energy of the 2D Incompressible Elastodynamics

Abstract:This talk concerns the time growth of the highest-order energy of the systems of two dimensional incompressible isotropic Hookean elastodynamics. This two dimensional system is nonlocal quasilinear wave equations where the unknowns has slow temporal decay. By observing an inherent strong null structure, the global well-posedness of smooth solutions near equilibrium was first proved by Lei where the highest-order generalized energy may have certain growth in time. We improve the result and show that the highest-order generalized energy is uniformly bounded for all the time.