Plenary Lecture 

• Juhi Jang (University of Southern California) 

Title : Dynamics of gaseous fluids near vacuum

Abstract : I will discuss recent progress on the dynamics of gaseous fluids governed by the compressible Euler and Euler-Poisson systems. In the first part, I will review the well-posedness and stability theory in the presence of physical vacuum free boundaries, and present recent joint work with Jiaqi Liu and Nader Masmoudi on self-similar waiting-time solutions exhibiting a transition from smooth to Hölder regularity and the emergence of moving vacuum boundaries. I will also discuss the construction of a large class of initially smooth, square-integrable solutions that develop gradient blowup near the boundary in finite time through a stability analysis of self-similar solutions. In the second part, I will discuss the dynamics of Newtonian stars near Lane-Emden equilibria for the Euler–Poisson system. While Lane-Emden stars between the mass-critical and energy-critical regimes are known to be nonlinearly unstable, their long-time nonlinear dynamics is yet to be understood. I will review recent progress and present upcoming joint work with Mahir Hadžić, Sung-Jin Oh, and Ely Sandine on a new instability mechanism: accelerated shock formation near the energy-critical Lane-Emden star.

Invited Talk

• Namhyun Eun (KIAS)

Title :  Stability for Oscillatory Shocks of KdV-Burgers Equation 

Abstract : The Korteweg-de Vries-Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks. These shocks are monotone in the viscosity-dominant regime, whereas they exhibit infinitely many oscillations in the dispersion-dominant regime.

In this talk, we study the stability of such viscous-dispersive shocks, focusing on an L^2 contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shock and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles.

This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).

• Younghun Hong (Chung-Ang University)

Title : Modified scattering for the Vlasov-Riesz system with long-range interactions

Abstract : We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential λ|x|^{−α} in the strictly long-range regime (0<α<1). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted W^{1,∞}-bounds. Compared with the earlier result, our Lagrangian approach extends modified scattering to the broader regime 1/2<α<1 and provides a distinct and more robust argument. This talk is based on joint work with Stephen Pankavich.

• Soyeun Jung (Kongju National University)

Title : Nonlinear instability of rolls in the two-dimensional generalized Swift-Hohenberg equation

Abstract : In this talk, we establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation. Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time.

• Hyangdong Park (KAIST)

Title : A Two-Phase Free Boundary Problem for Axisymmetric Subsonic Euler Flows with Contact Discontinuities

 Abstract : In this talk, we consider a genuinely two-phase free boundary problem for the three-dimensional steady compressible Euler equations in an infinitely long circular cylinder. The free boundary is a contact discontinuity separating an unknown rotational subsonic flow from an unknown potential subsonic flow. We show the existence of solutions and analyze their asymptotic behavior at downstream infinity. In particular, the contact discontinuity converges to a cylindrical configuration and the radial velocity decays to zero. This provides the first rigorous treatment of such a two-phase free boundary problem in a three-dimensional cylindrical setting.

• Wanyong Shim (KAIST)

Title : Asymptotic self-similar blow-up for the regularized Saint-Venant equations

Abstract : We study singularity formation in the regularized Saint-Venant (rSV) equations in one space dimension, a system that can be formally viewed as a Hamiltonian regularization of the isentropic Euler equations. Although smooth solutions to the rSV system are known to develop finite-time gradient blow-up, the precise structure of such singularities has not been rigorously characterized. In this talk, we present a sharp description of the asymptotically self-similar blow-up and establish C^{3/5} Hölder regularity at the singular time. This regularity differs from the C^{1/3} singularity observed in the isentropic Euler equations. This contrast motivates a discussion of the structural influence of the Hamiltonian regularization on singularity formation.

• Sihyun Song (Yonsei University)

Title : Some results on hypocoercivity

Abstract : The topic of hypocoercivity, which investigates the long time behavior of solutions to kinetic equations, is introduced. Basic toy examples such as ODEs will be used to motivate the theory, and some recent developments will then be summarized. Finally, some of my own recent/upcoming results on the long time behavior of nonlinear kinetic equations will be mentioned.