1st Workshops on Mathematical Fluid Dynamics

Date：October 29 - 31

Venue：E4-233 & E10-212, Yungu Campus

ZOOM ID: 991 7246 8807

Passcode: 257347

Ⅰ  Organizers

Alexey Cheskidov，Mimi Dai

Ⅱ  Agenda

Ⅲ  Talks

1. 10:00-11:00, October 29, Sunday

Speaker: De Huang, Peking University

Title: Exact self-similar finite-time blowups of some 1D models for the incompressible Euler equations.

Abstract: A series of 1D models have been proposed to study the competition between advection and vortex stretching of the 3D Euler equations. These models include the De Gregorio model, the generalized Constantin–Lax–Majda model, and the 1D Hou-Luo model. In this talk, we present some recent results on exact self-similar finite-time blowup solutions of these models. For the 1D De Gregorio model, we show that there exist infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions all correspond to the eigenfunctions of a self-adjoint compact operator. For the generalized Constantin–Lax–Majda model and the 1D Hou-Luo model, we establish existence of exact self-similar finite-time blowups using a novel fixed-point method. We also give detailed characterizations of these solutions on their regularities, monotonicity, and convexity.

2. 11:00-12:00, October 29, Sunday

Speaker: Weikui Ye, South China Normal University

Title: Some problems on the convex integral and non-uniqueness

Abstract: In this report, I mainly report to you that I have recently encountered several problems related to convex integrals, mainly include the 2D Onsager conjecture, the Compressible Euler system and the Compressible Navier-Stokes system.

3. 09:00-09:15, October 30, Monday

Speaker: Xiangyuan Li, Peking University

Title: A conjecture on the intrinsic nonlinear stability of De Gregorio-type equations.

Abstract: We study the nonlinear stability of equations that are structurally similar to the 1D De Gregorio equation. It has been proved that (1) the periodic De Gregorio equation is nonlinearly stable around the sine function, and (2) the dynamic rescaling equation of the De Gregorio equation on the real line is also nonlinearly stable around a compactly supported stationary solution. We will explain the essential similarity between these two stability results, which leads to our conjecture on the stable nature for a class of De Gregorio-like equations.

4. 09:15-09:30, October 30, Monday

Speaker: Xiao Ren, Peking University

Title: Stationary Navier-Stokes solutions in 2D exterior domains

Abstract: Consider the steady-state Navier-Stokes system in planar exterior domains. With no-slip boundary condition and prescribed limit velocity at infinity, this problem describes flows around cylindrical obstacles. An open problem since Leray (1933) asks whether solutions to this problem exist for arbitrary limit velocities. We present some recent progress concerning the uniqueness of solutions and the justification of Leray's method. We also discuss new results on the whole-plane forced problem. Based on joint works with M. Korobkov and J. Guillod.

5. 09:30-09:45, October 30, Monday

Speaker: Zhaojie Yang, Fudan University

Title: Energy Transfer and Radiation in Hamiltonian Nonlinear Klein-Gordon Equations

Abstract: We consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions. It is assumed that the corresponding Klein-Gordon operator admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein discovered a mechanism called Fermi's Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $\Omega \in (\frac{m}{3},m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. Since then, many efforts have been made in the case of relatively small eigenvalue, and the case of general multiple eigenvalue. In 2022, we solved the general one simple eigenvalue case. In our recent work, we solved this problem in full generality: multiple and simple or degenerate eigenvalues in $(0,m)$. Indeed, we obtained the sharp rate of energy transfer from discrete states to continuum modes in the most general case.

6. 09:45-10:00, October 30, Monday

Speaker: Cheng Yuan, Fudan University

Title: Backward uniqueness for 3D D Navier-Stokes Equations with Non-trivial Final Data.

Abstract: In this talk, I present a backward uniqueness result of bounded mild solutions of 3D Navier-Stokes Equations in the whole space with non-trivial final data. The proof is based on a new weighted estimate which enables to treat terms involving Calderon-Zygmund operators. The new weighted estimate is expected to have certain applications in control theory when classical Carleman-type inequality is not applicable. This is a joint work with Prof. Zhen Lei and Doct. Zhaojie Yang.

7. 10:00-11:00, October 30, Monday

Speaker: Xiaoqian Xu, Duke Kunshan

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 application 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.

8. 11:00-12:00, October 30, Monday

Speaker: Nie Yao, Nankai University

Title: Sharp ill-posedness for the non-resistive mhd equations in sobolev spaces

Abstract: In this talk, we show a sharp ill-posedness result for the incompressible non-resistive MHD equations. In any dimension $d\ge 2$, we show the ill-posedness of the non-resistive MHD equations in $H^{\frac{d}{2}-1}(\RR^d)\times H^{\frac{d}{2}}(\RR^d)$, which is sharp in view of the results of the local well-posedness in $H^{s-1}(\RR^d)\times H^{s}(\RR^d)(s>\frac{d}{2})$ established by  Fefferman et al.(Arch. Ration. Mech. Anal., \textbf{223} (2), 677-691, 2017). Furthermore, we generalize the ill-posedness results from $H^{\frac{d}{2}-1}(\RR^d)\times H^{\frac{d}{2}}(\RR^d)$ to Besov spaces $B^{\frac{d}{p}-1}_{p, q}(\RR^d)\times B^{\frac{d}{p}}_{p, q}(\RR^d)$ and $\dot B^{\frac{d}{p}-1}_{p, q}(\RR^d)\times \dot B^{\frac{d}{p}}_{p, q}(\RR^d)$ for $1\le p\le\infty, q>1$.  Different from the ill-posedness mechanism of the incompressible Navier-Stokes equations in $\dot B^{-1}_{\infty, q}$ \cite{B,W}, we construct an initial data such that  the  paraproduct terms (low-high frequency interaction) of the nonlinear term  make the main contribution to the norm inflation of the magnetic field. This work is joint with Qionglei Chen and Weikui Ye.

9. 9:00-10:00, October 31, Tuesday

Speaker: Fei Wang, Shanghai Jiao Tong University

Title: On the Euler+Prandtl expansion for the Navier-Stokes equations

Abstract: We establish the validity of the Euler+Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an L 1 -type norm for the vorticity of the error term in the expansion Navier-Stokes-(Euler+Prandtl). An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.

10. 10:00-11:00, October 31, Tuesday

Speaker: Xiaojing Xu, Beijing Normal University

Title: Some Problems on global well-posedness of the 2D Boussinesq Equations

Abstract: In this talk, I will first introduce some known results on global well-posedness of the Cauchy problem of 2D Boussinesq Equations with some kinds of dissipative terms, then I will give some unsovled problems on the 2D Boussinesq equations.

11. 11:00-12:00, October 31, Tuesday

Speaker: Yifei Wu, Tianjin University

Title: Non-relativistic limit for the cubic nonlinear Klein-Gordon equations

Abstract: In this talk, focuses on the non-relativistic limit of the Cauchy problem for the defocusing cubic nonlinear Klein-Gordon equations. We show that, as the light speed $c$ tends to infinity, the error function is bounded by, (1) in the case of 2D and modulated Schrodinger-wave profiles, $c^{-2}$, uniformly for all time, under $H^2$ initial data; (2) in the case of both 2D and 3D and modulated Schr\"odinger profiles, $c^{-2} +(c^{-2}t)^{\alpha/4}$, under $H^\alpha$ initial data with $2 \leq \alpha \leq 4$. We also show the sharpness of the upper bounds in (1) and (2), and the required minimal regularity on the initial data in (2). This talk is based on a joint work with Zhen Lei.