Invited Speakers

Organizers/contact

Daomin Cao (AMSS, CAS)

Guolin Qin (AMSS, CAS) Email: qinguolin18@mails.ucas.ac.cn

Venue: N202, South Building, Academy of Mathematics and Systems Science, CAS, 55 Zhongguancun East Road, Haidian District, Beijing, China

Welcome to join!

1. Schedule

1. Abstracts

Speaker：Ken Abe (Osaka Metropolitan University)

Title: Homogeneous steady states for the generalized surface quasi-geostrophic equations

Abstract: I will discuss spatially homogeneous steady states for the gSQG equations in the whole plane. Those homogeneous solutions are stationary self-similar/scale-invariant solutions in the time-dependent problem. I will discuss a variational approach to construct such homogeneous solutions for some triplets (s,\beta,m): (i) 0<s<1; the parameter for the constitutive laws, (ii) \beta; homogeneity acting on the radial variable, and (iii) m; frequency for the angular variable. This talk is based on joint work with J. Gomez-Serrano (Brown U.) and I.-J. Jeong (KIAS).

Speaker：Kyudong Choi (Ulsan National Institute of Science and Technology)

Title: Vortex Atmospheres: 2D Dipoles versus 3D Rings

Abstract: We consider traveling solutions in the incompressible Euler equations in the vorticity form in two or three dimensions. The support of such a traveling solution is called the core of the vortex. The vortex atmosphere is an envelope of the core, which is traveling together with the core. In this presentation, we offer a concise survey of vortex atmosphere configurations. By rigorous analysis, we contrast the planar atmospheres from 2D vortex dipoles with the axisymmetric atmospheres from 3D vortex rings, highlighting their topological differences. This is joint work with In-Jee Jeong (KIAS) and Young-Jin Sim (UNIST).

Speaker：In-Jee Jeong (Korea Institute for Advanced Study)

Title:Stability of traveling waves on the half-plane without sign condition on the vorticity

Abstract: For the 2D incompressible Euler equations on the half-plane, we establish Lyapunov stability of the Lamb dipole without sign condition on the vorticity. This seems to be the first stability result for traveling waves which does not require sign condition in the half-plane. We present several extensions and applications of this result. Joint work with Ken Abe, Kyudong Choi, and Guolin Qin.

Speaker：Deokwoo Lim (Seoul National University)

Title:On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

Abstract: We consider incompressible Euler flows with axisymmetry and without swirl. In $\mathbb{R}^{3}$, we prove the $t^{4/3}$-upper bound for the growth of the vorticity maximum. This was conjectured by Childress (Phys. D, 2008) and supported by numerical computations from Childress—Gilbert—Valiant (J. Fluid Mech., 2016). The key idea is to estimate the velocity maximum by the kinetic energy and conserved quantities related to the vorticity. This is a joint work with In-Jee Jeong (SNU).

Speaker：Jinmyoung Seok (Seoul National University)

Title: Dynamical stability of steady states in Vlasov-Poisson and Euler-Poisson equations

Abstract: In this talk, I will present some unified ideas for constructing steady states and establishing their nonlinear dynamical stability for Vlasov and Euler-Poisson equations. The first part will be devoted to foundational theory and key historical developments. In the second part, I will discuss several recent results, including some of my own work.

Speaker：Young-Jin Sim (Korea Institute for Advanced Study)

Title:Stability of two Hill's vortices and their speed estimate

Abstract:We establish the stability of a pair of Hill's spherical vortices moving away from each other in 3D incompressible axisymmetric Euler equations without swirl. Each vortex in the pair propagates away from its odd-symmetric counterpart, while keeping its vortex profile close to Hill's vortex. This is achieved by analyzing the evolution of the interaction energy of the pair and combining it with the compactness of energy-maximizing sequences in the variational problem concerning Hill's vortex. The key strategy is to confirm that, if the interaction energy is initially small enough, the kinetic energy of each vortex in the pair remains so close to that of a single Hill's vortex for all time that each vortex profile stays close to the energy maximizer: Hill's vortex. An estimate of the propagating speed of each vortex in the pair is also obtained by tracking the center of mass of each vortex. This estimate is optimal in the sense that the power exponent of the epsilon (the small perturbation measured in the "L^1+L^2+impulse" norm) appearing in the error bound cannot be improved. This talk is based on the paper [Y.-J. Sim, Nonlinearity, 2026].

Speaker：Guodong Wang (Dalian University of Technology)

Title:Stability of first Laplacian eigenstates for the Euler equation on a flat 2-torus

Abstract:On a flat 2-torus, the Laplacian eigenfunctions can be expressed in terms of sinusoidal functions. For a rectangular or square torus, it is known that every first eigenstate is orbitally stable up to translation under the Euler dynamics. In this talk, we show that this is also true for flat tori of arbitrary shape. As a corollary, we obtain for the first time a family of orbitally stable sinusoidal Euler flows on a hexagonal torus.

Speaker：Kwan Woo (University of Basel)

Title: Existence of smooth Sadovskii vortex

Abstract: I will present a variational construction of smooth Sadovskii vortices for the two-dimensional incompressible Euler equations. The Lamb dipole, as the most classical example of a Sadovskii vortex, is Lipschitz, and it is then natural to ask whether one can construct axis-touching traveling dipoles with higher regularity. I will explain a recent result that overcomes the C^3 regularity barrier of previous methods. This is joint work with Abe, Choi, Jeong, and Sim.

Speaker：Yao Yao (National University of Singapore)

Title: Infinite-in-time growth in 3D incompressible Euler equations without swirl

Abstract: In this talk, I will discuss some infinite-in-time growth results for the 3D axisymmetric Euler equation without swirl. Namely, we establish some upper and lower bound for the radial moment of vorticity, and prove that under some sign and symmetry conditions, all solutions must have their vorticity L^p norm growing to infinity with some power-law rate for all p>=1 (joint with Khakim Egamberganov).

Speaker：Weicheng Zhan (Xiamen University)

Title: On the radial symmetry of stationary and uniformly rotating solutions of the 2D Euler equation

Abstract: In this talk, I will present several rigidity results for relative equilibria of the two-dimensional incompressible Euler equations, with a focus on the radial symmetry of stationary and uniformly rotating solutions in both the whole plane and the unit disk. We consider both smooth solutions and vortex patches, illustrating how the interplay between vorticity and angular velocity leads to rigidity phenomena. In particular, for a given rotating solution $\omega_0$, we show that it must be radially symmetric whenever the angular velocity satisfies $\Omega \le \inf \omega_0/2$ or $\Omega \ge \sup \omega_0/2$. A key ingredient in our analysis is the symmetry of nonnegative solutions to elliptic equations. To tackle the symmetry problem for nonnegative solutions of piecewise-coupled semilinear elliptic equations, we develop a novel method specifically designed for this setting. This talk is based on joint work with Boquan Fan and Yuchen Wang.

Speaker：Tao Zhou (National University of Singapore)

Title:Superlinear gradient growth for 2D Euler equation without boundary

Abstract:We consider the vorticity gradient growth of solutions to the two-dimensional Euler equations in domains without boundary, namely in the torus \mathbb{T}^{2} and the whole plane \mathbb{R}^{2}. In the torus, whenever we have a steady state \omega^* that is orbitally stable up to a translation and has a saddle point, we construct \tilde\omega_0 \in C^\infty(\mathbb{T}^2) that is arbitrarily close to \omega^* in L^2, such that superlinear growth of the vorticity gradient occurs for an open set of smooth initial data around \tilde\omega_0. This seems to be the first superlinear growth result which holds for an open set of smooth initial data (and does not require any symmetry assumptions on the initial vorticity). Furthermore, we obtain the first superlinear growth result for smooth and compactly supported vorticity in the plane, using perturbations of the Lamb--Chaplygin dipole.This talk is based on the joint talk with In-Jee Jeong and Yao Yao.

Speaker：Changjun Zou (Sichuan University)

Title:Uniqueness and nonlinear orbital stability of steady vortex rings of small cross-section

Abstract: We will talk about steady vortex rings in an ideal fluid of uniform density, which are special global axi-symmetric solutions to the three-dimensional incompressible Euler equations. In particular, we consider the case where the cross-section is of small epsilon-scale and the potential vorticity is constant throughout the core. We will first introduce the existence results via different methods. Then we will discuss the uniqueness of such thin vortex rings by detailed expansion of stream function and the local Pohozaev identity. Finally, we will give the proof for the nonlinear orbital stability by maximum energy properties of vortex rings and applying Arnol′d’s variational principle. These results answer a long-standing question posed since the pioneering work of Fraenkel and Berger (Acta Math., 1974).