Plenary Lecture 

• Hantaek Bae (Ulsan Natinoal Institute of Science and Technology)

Title : Mathematical Analysis of MHD Part 1, 2

Abstract : 

Invited Talk

• Sukjung Hwang (Chungbuk National University)

Title :  Local Boundedness for Parabolic p-Laplace Equations via De Giorgi Iteration

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).

• Hyunjin In (Ajou University)

Title : Liouville-Type Theorems for Three-Dimensional Stationary Compressible Flows

Abstract : We discuss Liouville-type theorems for stationary compressible fluid equations in \R^3. We first review Liouville-type results for the incompressible stationary Navier–Stokes equations, with an emphasis on pressure estimates. We then compare pressure estimates for the incompressible and compressible stationary Navier–Stokes equations and introduce an estimate adapted to the compressible case. Finally, we consider the stationary ideal compressible magnetohydrodynamics equations, focusing on their connection to Beltrami flows. This talk is based on joint work with Youseung Cho (Yonsei University), Sarka Necasova (Czech Academy of Sciences), and Minsuk Yang (Yonsei University).

• Junha Kim (Ajou University)

Title : Boundary effects for classical solutions of \alpha-SQG equations in the half-plane

Abstract : We investigate the effect of the boundary of the half-plane on the velocity and identify, for smooth initial data, the threshold regularity for local wellposedness. First, we consider data that do not vanish on the boundary. Since the dynamics in the half-plane corresponds to the odd extension of the data to the whole plane, the extended data have a jump discontinuity across the boundary, and the Biot--Savart law indicates that the velocity is only $C^{1-\alpha}$. Decomposing the tangential velocity into a regular part and a singular part, we extract the exact form of the singular part, which shows that this H\"older order $1-\alpha$ is attained near boundary points where the data do not vanish. Exploiting this decomposition and the structure of the nonlinearity, we prove that the solution persists in an anisotropic H\"older space contained in $C^{1-\alpha}$ for $0<\alpha\le1/2$. This is sharp: smooth data immediately lose $C^\beta$ regularity for every $\beta>1-\alpha$, and for $1/2<\alpha\le1$ there is no $C^\alpha$ solution at all. Second, we turn to the inviscid SQG equation with the Dirichlet boundary condition. Capturing the logarithmic singularity of the second normal derivative of the tangential velocity generated by the boundary, and combining it with the structure of the nonlinearity, we show that $C^{2,1}$ is the threshold: solutions with $C^\infty_c$ data vanishing on the boundary remain in $C^{2,\beta}$ for a short time for every $0<\beta<1$, while under an explicit condition on the data they lose $C^{2,1}$ regularity instantly. In both cases, the loss of regularity occurs in the normal derivatives of the solution at the boundary. This talk is based on joint works with Prof. In-Jee Jeong, Prof. Yao Yao, and Prof. Hideyuki Miura.

• Chanhong Min (Yonsei University)

Title : 

 Abstract : 

• Jung-Tae Park (Korea University of Technology and Education)

Title :  Nonlinear diffusion equations with drift and measure data

Abstract :