Date: May 30 - June 1

Venue: E4-233, Yungu Campus

Zoom ID: 946 8893 9106

Passcode: 715681

Ⅰ  Agenda

Date	Time	Venue	Title	Speaker
5.30	9:00-10:00	E4-233	Small scale creation in equations of fluid mechanics (Lecture Series)	Alexander Kiselev
	10:00-11:00		Norm inflation for Navier-Stokes equations in supercritical spaces	Xiaoyutao Luo
	11:00-12:00		TBA	TBA
	12:00-13:00		Lunch
	13:30-14:30	E5-329	Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth Models	Jiajun Tong
	14:30-15:30		Mixing flow and advection-diffusion-reaction equations	Xiaoqian Xu
	15:30-17:30		Discussion
	17:30		Dinner
5.31	9:00-10:00	E4-233	Suppression of chemotactic blow up by buoyancy (Lecture Series)	Alexander Kiselev
	10:00-11:00		The Relative Entropy Method in the Mean Field Limit	Zhenfu Wang
	11:00-12:00		Multi-bubble blow-ups and multi-solitons to (stochastic) nonlinear Schrödinger equations	Deng Zhang
	12:00-13:00		Lunch
	13:30-14:30		On self-similar finite-time blowups of the incompressible Euler equations and related models	De Huang
	14:30-15:30		A geometric characterization of potential Navier-Stokes singularities	Xiao Ren
	15:30-16:30		Regularity of vortex and SQG patches (Lecture Series)	Alexander Kiselev
	16:30-17:30		Discussion
	17:30		Dinner
6.1	9:00-10:00	E4-233	The flow of polynomial roots under differentiation (Lecture Series)	Alexander Kiselev
	10:00-11:00		Dissipative solutions of the 3D Naiver-Stokes equations	Zirong Zeng
	11:00-11:20		Global well-posedness of the Navier--Stokes equations for small initial data	Taichi Eguchi
	11:20-11:40		Finite time blowup for Keller-Segel equation with logistic damping in three dimensions	Tao Zhou
	11:40-12:00		Vorticity gradient growth for incompressible Euler equations in certain unbounded domains	Khakimboy Egamberganov
	12:00-13:00		Lunch
	13:30-17:30		Discussion
	17:30		Dinner

Ⅱ  Talks

May 30, Friday

9:00-10:00

Speaker: Alexander Kiselev, Duke Univeristy

Title: Small scale creation in equations of fluid mechanics (Lecture Series)

Abstract: Most of the fundamental equations of fluid mechanics are nonlinear and nonlocal. Some of the key examples are the Euler equation for ideal fluid, the surface quasi-geostrophic (SQG) equation coming from atmospheric science, the Boussinesq system modeling buoyant fluid, and the incompressible porous media (IPM) equation. Solutions to these equations are often prone to instabilities, small scale creation and possibly singularity formation. I will overview some of the recent results on small scale creation in solutions of fluid mechanics, with the focus on recent works on the porous media equation and Boussinesq system joint with Benjamin Park and Yao Yao.

10:00-11:00

Speaker: Xiaoyutao Luo, Chinese Academy of Sciences

Title: Norm inflation for Navier-Stokes equations in supercritical spaces

Abstract: We will show that the incompressible 3D Navier-Stokes equations exhibit norm inflations in almost all supercritical spaces.

11:00-12:00

Speaker: TBA

Title: TBA

Abstract: TBA

13:30-14:30

Speaker: Jiajun Tong, Peking University

Title: Convergence of Free Boundaries in the Incompressible Limit of Tumor Growth Models

Abstract: We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this talk, we shall present an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. It relies on quantifying the relation between the free boundary motion and spatial average of the pressure, and establishing a uniform-in-m strict expansion property of the pressure supports. We derive upper bounds for the Hausdorff dimensions of the free boundaries. This is based on a joint work with Yuming Paul Zhang (Auburn University).

14:30-15:30

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. I will also discuss about the examples of such flows in this talk.

May 31, Saturday

9:00-10:00

Speaker: Alexander Kiselev, Duke Univeristy

Title: Suppression of chemotactic blow up by buoyancy (Lecture Series)

Abstract: There exist many regularization mechanisms in nonlinear PDE that help  make solutions more regular or prevent formation of singularity: diffusion, dispersion, damping. A relatively less understood regularization mechanism is transport. There is evidence that in the fundamental PDE of fluid mechanics such as Euler or Navier-Stokes, transport can play a regularizing role. In this talk, I will discuss  another instance where this phenomenon appears: the Patlak-Keler-Segel equation of chemotaxis. Chemotactic blow up in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of a given fluid advection can arrest singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong. This talk will focus on the case when the fluid advection is active: the Patlak-Keller-Segel equation coupled with fluid that obeys Darcy's law for incompressible porous media flow via gravity. Surprisingly, in this context, in contrast with the passive advection, active fluid is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions. The talk is based on works joint with Zhongtian Hu, Naji Sarsam, and Yao Yao.

10:00-11:00

Speaker: Zhenfu Wang, Beijing International Center for Mathematical Research (BICMR) , Peking University

Title: The Relative Entropy Method in the Mean Field Limit

Abstract: In this talk, I will review recent progress in the mean field limit and the propagation of chaos for large systems of interacting particles, including cases with singular interactions, using the relative entropy method. In particular, we will focus on the whole-space setting, highlighting key examples such as particle approximations of the two-dimensional Navier-Stokes equations and Landau equations with Maxwellian molecules in arbitrary dimensions. To achieve these results, we develop essential pointwise logarithmic gradient and Hessian estimates of the density function, inspired by the Li-Yau approach. This presentation is based on a series of recent collaborations with X. Feng (Peking University), J.A. Carrillo (University of Oxford), S. Guo (University of Oxford), and P.-E. Jabin (Penn State University).

11:00-12:00

Speaker: Deng Zhang, Shanghai Jiao Tong University

Title: Multi-bubble blow-ups and multi-solitons to (stochastic) nonlinear Schrödinger equations

Abstract: In this talk I will review some results on multi-bubble blow-ups and multi-solitons to the focusing mass-critical nonlinear Schrödinger equations (NLS) in both stochastic and deterministic cases. We will show the construction and conditional uniqueness of multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. The proof strategy also applies to the construction of multi-bubble blow-ups for the mass-critical half-wave equation. Furthermore, in the low asymptotic regime, the refined uniqueness is derived for the deterministic NLS.

13:30-14:30

Speaker: De Huang, Peking University

Title: On self-similar finite-time blowups of the incompressible Euler equations and related models.

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. In this talk, I will review some most recent results on finite-time blowups with self-similar features, divided into three parts. In the first part, I will talk about the dynamic rescaling method for establishing asymptotically self-similar finite-time blowup with rigorous computer-assisted proof. In the second part, I will talk about recent findings on potential self-similar finite-time blowups of the 3D Euler equations with multi-scale features, which are closely related to traveling wave solutions and may provide a new approach towards Euler singularity. In the last part, I will introduce our construction of Sadovskii’s vortex-patch dipole as a traveling wave solution of 2D Euler by a fixed-point method.

13:30-14:30

Speaker: Xiao Ren, Peking University

Title: A geometric characterization of potential Navier-Stokes singularities

Abstract: In 1993, Constantin and Fefferman proved that spatial Lipschitz continuity of vorticity directions precludes singularity formation, establishing an interesting connection between vortex alignment and regularity. Extending this paradigm, we investigate a scale-invariant scenario where vorticity remains confined within a double cone. We prove that such solutions stay regular - in other words, near any potential singularity, the vorticity direction set has to touch every great circle on the unit sphere.The proof is inspired by the Kelvin-Helmholz law for ideal fluids and the critical regularity theory for axisymmetric solutions. Based on joint work with Zhen Lei and Gang Tian.

13:30-14:30

Speaker: Alexander Kiselev, Duke Univeristy

Title: Regularity of vortex and SQG patches (Lecture Series)

Abstract: Patch solutions are an important class of special singular solutions to the 2D Euler or surface quasi-geostrophic (SQG) equations that model evolution of regions of vorticity with sharp boundaries (like hurricanes) or sharp temperature fronts in atmosphere. I will discuss recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to C^2 class at all integer times only without being time periodic. The proof involves derivation of a new system describing the patch evolution in terms of arc-length and curvature. A similar approach leads to discovery of strong ill-posedness of the SQG patches in all but L^2 based spaces, or spaces of infinitely smooth functions. The talk is based on a work joint with Xiaoyutao Luo.

June 1, Sunday

9:00-10:00

Speaker: Alexander Kiselev, Duke Univeristy

Title: The flow of polynomial roots under differentiation (Lecture Series)

Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of polynomial roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species - such as fish, birds or ants. I will discuss joint work with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.

10:00-11:00

Speaker: Zirong Zeng, Nanjing University of Aeronautics and Astronautics

Title: Dissipative solutions of the 3D Naiver-Stokes equations

Abstract: In this talk, I will present our recently result about dissipative solutions of the 3D Naiver-Stokes equations. For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term. This talk is based on the joint work with Alexey Cheskidov and Deng Zhang.

11:00-11:20

Speaker: Taichi Eguchi, Waseda University

Title: Global well-posedness of the Navier--Stokes equations for small initial data

Abstract: We construct a global-in-time smooth solution to the Navier--Stokes equations for small initial data. The initial data belongs to $L^2$, and is split into two parts. One is the higher frequency part than a large number, and the other is the low-frequency part, which is the remaining part of the high-frequency part. We assume that the  high-frequency part is small in $BMO^{-1}$, and the low-frequency part is small only in $\dot B^{-1}_{\infty,\infty}$. Hence, our smallness assumption is weaker than Koch--Tataru (2001). Moreover, our solutions attain the initial data in the strong $L^2$ sense and satisfy the energy equality for all time. This is a joint work with Prof. A. Cheskidov.

11:20-11:40

Speaker: Tao Zhou, National University of Singapore

Title: Finite time blowup for Keller-Segel equation with logistic damping in three dimensions

Abstract: The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-\mu \rho^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $\mu \in \big[0,\frac{1}{3}\big)$. This range of $\mu$ is sharp; for $\mu \ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation. This talk is based on the joint work with Jiaqi Liu (USC) and Yixuan Wang (Caltech).

11:40-12:00

Speaker: Khakimboy Egamberganov, National University of Singapore

Title: Vorticity gradient growth for incompressible Euler equations in certain unbounded domains

Abstract: We investigate the growth of the vorticity gradient for incompressible Euler equations within certain unbounded domains. We construct initial data for both the 2D Euler equations and the axisymmetric 3D Euler equations without swirl in unbounded smooth symmetric domains that leads to double exponential growth of the vorticity gradient over time.