Plenary Lecture 

• In-Jee Jeong (Seoul National Univ.) 

Title : Long time dynamics of incompressible Euler equations on the plane

Abstract : In the first part, we consider conserved quantities of the incompressible Euler equations on the plane, and the monotone quantities under the odd symmetry and vorticity sign condition. Then, we introduce the variational principle which selects the Lamb dipole as the unique energy maximizer. We explain how dynamical stability of the Lamb dipole follows from the variational principle.

In the second part, we introduce the Lagrangian approach for the incompressible Euler equations, which tracks locations of the fluid particles in time. Then we explain how dynamical stability of the Lamb dipole can be combined with the Lagrangian argument to prove occurrence of filamentation for perturbations of the Lamb dipole.

Invited Talk

• Jae-Hwan Choi (KIAS) 

Title : Inviscid limit for the two-dimensional Navier–Stokes equation: Stochastic Lagrangian approach

Abstract : I will discuss recent progress on the vanishing-viscosity limit of the two-dimensional Navier–Stokes equation. Our approach is Lagrangian and probabilistic:

1. We develop a stochastic counterpart of the DiPerna–Lions theory to construct and control stochastic Lagrangian flows for the viscous dynamics.

2. We also establish a large-deviation principle that quantifies convergence to the Euler dynamics.

This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.

• Sungho Han (KAIST) 

Title : Stability of Viscous-Dispersive Shock Waves for the Navier-Stokes-Korteweg System

Abstract : In this talk, I will discuss the long-time behavior of the one-dimensional barotropic Navier-Stokes-Korteweg system. For small-amplitude Riemann data generating a viscous–dispersive shock profile, we show that solutions converge to this profile up to a time-dependent shift. The analysis is based on an extension of the a-contraction with shift method, adapted to handle the third-order Korteweg term. We also treat Riemann data whose inviscid solution contains a rarefaction wave and a shock wave, and prove the time-asymptotic stability of the associated viscous–dispersive composite wave in the small-amplitude regime. These results are based on joint works

[1] S. Han, M.–J. Kang, J. Kim, and H. Lee, Long-time behavior toward viscous–dispersive shock for Navier–Stokes equations of Korteweg type, J. Differential Equations 426 (2025), 317–387.

[2] S. Han and J. Kim, Time-asymptotic stability of composite waves for the one-dimensional compressible fluid of Korteweg type, to appear in SIAM Journal on Mathematical Analysis.

• Dong-ha Kim (Ajou Univ.) 

Title : Critical threshold and large-time behavior in 1D presureless Euler-Poisson system

Abstract : In this talk, we introduce the one-dimensional pressureless Euler-Poisson system with background states.

First, we will discuss the well-posedness and ill-posedness of the system, which includes a rigorous treatment of the neutrality condition. We then consider critical threshold phenomena which deternmines global-in-time existence and finite-time singularity formation. Lastly, for the damped case, assuming a global-in-time regularity, we will see the large-time behavior (or asymptotic behavior) of this solution. As an application of the repulsive case, we study the case of cold-plasma problem.

This is joint work with Young-pil Choi, Dowan Koo, Eitan Tadmor.

• Joonhyun La (KIAS) 

Title : Wave turbulence and some applications

Abstract : In this talk, we briefly introduce wave turbulence theory, a statistical theory of nonlinear dispersive systems in weakly turbulent regimes. Then we see applications of wave turbulence theory - on MMT equation, an one-dimensional toy model to understand wave turbulence, and on FPUT-experiment. The talk is based on a joint work with Pierre Germain (ICL) and Zhiyuan Zhang (Northeastern), and one with Pierre Germain and Angeliki Menegaki (ICL).

• Chanhong Min (Yonsei Univ.) 

Title : Asymptotic behavior of the normal coordinate of vanishing velocity of Stokes equations in the half space

Abstract : We consider the unsteady Stokes system in the half–space with zero initial data and nonzero, space–time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign away from the boundary. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half–space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.

• Young-Jin Sim (UNIST) 

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 can be understood as 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 preprint [arXiv:2507.19935].