Special Session 143: Nonlinear dynamics for kinetic, fluids and mathematical physics

Decay Rates of Solutions to Some Dissipative Equations

Hantaek Bae UNIST Korea Co-Author(s):

Abstract: In this talk, we demonstrate how to derive temporal decay rates for weak solutions to the incompressible Navier-Stokes equations. We establish these rates by employing the Fourier splitting method with initial data in spaces of negative regularity. We also discuss applications of our approach to the surface quasi-geostrophic (SQG) equations and magnetohydrodynamics (MHD).

Existence for steady-state $p(u)$-Laplacian problems related with Image Processing

Hyeong-Ohk Bae Ajou University Korea Co-Author(s):

Abstract: We construct weak solutions to the $p(u)$--Laplace equation $- \nabla\cdot \big(|\nabla u|^{p(u) -2} \nabla u\big) = f \quad \text{in } \Omega.$ These problems arise in adaptive image denoising, where the diffusion depends on the solution itself. The main challenge stems from the variable exponent, which prevents the use of fixed Sobolev spaces. We overcome this difficulty by employing a sophisticated version of the Lipschitz truncation method, combined with a localized Minty argument. This approach enables us to prove the existence of weak solutions under minimal regularity assumptions, without requiring the restriction $p(u) > d$, where $d$ is the space dimension. Our result applies to a broader class of nonlinear diffusion equations and contributes to the analysis of models where the equation`s behavior may depend on the solution itself.

Analysis of Voigt-type fluid models: existence, regularity and long-time behavior

Hermenegildo B de Oliveira University of the Algarve Portugal Co-Author(s):

Abstract: In this talk we present several recent results on the mathematical analysis of Voigt-type regularizations of fluid flow models. These models extend the classical Navier-Stokes framework by incorporating relaxation or viscoelastic effects, leading to improved analytical properties while preserving the main features of the fluid dynamics models. We study different formulations of these systems, including generalized power-law models, flows with non-homogeneous density, and models allowing the presence of vacuum regions. For the associated nonlinear initial-boundary value problems, we establish results on the existence, uniqueness, and regularity of weak and strong solutions under suitable assumptions on the nonlinear exponents and the spatial dimension. We also investigate qualitative properties of the solutions, including long-time behavior and decay rates in the presence of source or sink terms, as well as smoothing and decay mechanisms for the linearized barotropic Navier-Stokes-Voigt system. These results contribute to a unified mathematical framework for the analysis of Voigt-regularized fluid models.

Convergence and aymptotics for multi-component Ginzburg-Landau equations

Jongmin Han Kyung Hee University Korea Co-Author(s):

Abstract: Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial \Omega$ and of degree zero. According to [Bethuel-Brezis-H\`{e}lein, 1993], the $H^1$ convergence of $u_\varepsilon$ is a key ingredient of uniform convergence. In this talk, we give an equivalent condition for $H^1$ convergence. Furthermore, we generalize it to multi-component Ginzburg-Landau equations. We also talk about the recent progress of asymptotics of minimizers for the multi-component Ginzburg-Landau energy.

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

Younghun Hong Chung-Ang university Korea Co-Author(s):    Younghun Hong; Stephen Pankavich

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 $\lambda |x|^{-\alpha}$ in the strictly long-range regime ($0 < \alpha < 1$). 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 $W^{1,\infty}$-bounds. Compared with the earlier result \cite{HuangKwon}, our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}

Optimal Convergence Estimate of the Limit from Inverse Power Potential to Hard Sphere Boltzmann Equation

Jin Woo Jang POSTECH (Pohang University of Science and Technology) Korea Co-Author(s):    Zheng-Nan Hu, Jin Woo Jang, Zheng-An Yao, Yu-Long Zhou

Abstract: The inverse power potential $U(r)=r^{-1/s}$, $0\lt s\lt 1$, generates the Boltzmann kernel $B^{s}=|v-v_*|^{1-4s} b_s(\theta)$ with an angular singularity as $\theta\to 0$. Jang et~al.~\cite{Jang2023-df} proved the limit $B^{s}\to \frac14|v-v_*|$ as $s\to 0$, as well as weak convergence of solutions based on this kernel convergence. In this talk I further introduce recent establishment of the following sharp quantitative estimate: \[ |b_s(\theta)-\tfrac14| \le C\, s\,\theta^{-2-2s}. \] In particular, this sharp estimate yields the \emph{optimal} $O(s)$ convergence rate for solutions of the homogeneous Boltzmann equation with large initial data in suitable Sobolev spaces; i.e., for any $t\in[0,T]$, we have \[ f^s(t)=f^0(t)+O(s), \] quantified by the $L^1_k$ norm for $k\ge 2$. This is a joint work with Zheng-Nan Hu, Zheng-An Yao, and Yu-Long Zhou.

Maximal $L^p$ Regularity for Nonhomogeneous Neumann Problems and Its Applications to Fluid Dynamics

Bum Ja Jin Mokpo National University Korea Co-Author(s):

Abstract: This talk presents the maximal $L^p$ regularity for the heat equation subject to nonhomogeneous Neumann boundary conditions and a divergence-form external source. While maximal regularity is well-established for homogeneous cases, nonhomogeneous boundary data often introduce technical complexities, particularly in fluid dynamics models. Our work establishes a robust maximal $L^p$ estimate for the generalized solution to this problem. As a key application, we demonstrate how this estimate improves the weak solution framework for the compressible Navier-Stokes equations with inflow-outflow boundary conditions. Specifically, we show that our regularity results allow for the removal of the artificial terms previously required in approximate systems (e.g., [Chang, Jin, and Novotny]), thereby providing a more physical and streamlined analysis of the Navier-Stokes equations under nonhomogeneous boundary settings.

Stability of compressible flows with boundary in one space dimension

Moon-Jin Kang Korea Advanced Institute of Science and Technology Korea Co-Author(s):

Abstract: The stability of entropy solutions to the compressible Euler equations on a halfline with prescribed boundary conditions (such as inflow; outflow; impermeable problems) remains a largely open problem. For the Euler system as hyperbolic equations, the dimension of admissible boundary set depends on the characteristic speeds, making it highly nontrivial to determine under which boundary set the stability can be guaranteed. Even the fundamental questions on how a single Riemann shock approaching the boundary behaves and whether such a shock is stable remains open. In this talk, I will present a recent progress concerning these problems.

On wellposedness of alpha-SQG equations in the half-plane

Junha Kim Ajou University Korea Co-Author(s):    In-Jee Jeong, Yao Yao

Abstract: We investigate the wellposedness of alpha-SQG equations in the half-plane, where alpha=0 and alpha=1 correspond to the 2D Euler and SQG equations respectively. We prove local wellposednesss and illposedness in anisotropic weighted Holder spaces. This talk is based on joint work with In-Jee Jeong and Yao Yao.

On the Role of Energy Stability in Numerical Methods for Incompressible Fluid Equations

Byungjoon Lee The Catholic University of Korea Korea Co-Author(s):    Chohong Min, Jeongho Kim, Dongnam Ko

Abstract: Energy stability plays a fundamental role in the design of reliable numerical methods for incompressible flow problems. In this talk, we present a unified framework for constructing energy-stable schemes across a range of challenging settings, including variable-density Navier-Stokes flows, high-order time discretizations, two-phase flows with surface tension, and fluid-solid interaction. Our approach is based on discrete energy laws that guarantee non-increasing total mechanical energy at the fully discrete level. Key ingredients include reformulations of the governing equations using suitable state variables, structure-preserving discretizations that mimic integration-by-parts on staggered grids, and both implicit and explicit time integration strategies with provable stability properties. In particular, we develop schemes that are conditionally or unconditionally energy stable while achieving improved accuracy, including second-order convergence. We further extend these ideas to adaptive grid frameworks for complex geometries and multiscale interactions, where consistent projections across grid interfaces preserve stability. Numerical experiments demonstrate that the proposed methods achieve robustness and efficiency across diverse incompressible flow regimes, highlighting the central role of energy stability in modern computational fluid dynamics.

L\infty theory with stretched exponential weight for the Boltzmann equation

DONGHYUN LEE POSTECH Korea Co-Author(s):    Sungbin Park and Jongin Kim

Abstract: Even in the spatially homogeneous case, any Maxwellian upper bound for the Boltzmann solution must be characterized by an exceptionally small decay rate in the velocity variable. This motivates us to study the Boltzmann equation with a stretched exponential weight, $e^{-\alpha|v|^\kappa}$ for $\kappa < 2$. In this talk, we discuss the well-posedness of the Boltzmann equation in $L^\infty_{x,v}$ within this stretched exponential framework. In particular, we address the large-amplitude problem of the Boltzmann equation and beyond.

Homogeneous solutions to the Einstein--Boltzmann system with a conformal gauge singularity

Ho Lee Kyung Hee University Korea Co-Author(s):    Ernesto Nungesser, John Stalker, Paul Tod

Abstract: We consider the Einstein--Boltzmann system for massless/massive particles in the Bianchi I spacetime with scattering cross-sections in a certain range of soft potentials. We assume that the spacetime has an initial conformal gauge singularity and show that the initial value problem is well posed with data given at the singularity.

A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed Neural Networks

Myeong-Su Lee Seoul National University Korea Co-Author(s):    Gyounghun Ko, Sung-Jun Son, and Seung-Yeon Cho

Abstract: While Physics-Informed Neural Networks offer a promising framework for partial differential equations, the standard L2 residual loss is insufficient for solving the Bhatnagar-Gross-Krook model of the Boltzmann equation. Specifically, small L2 residuals do not guarantee small errors in macroscopic moments, causing the standard PINN to fail in capturing the true solution. To resolve this, we propose a weighted L2 loss function based on a stability analysis. The stability analysis implies that minimizing this weighted residual strictly controls the distance between the exact and approximate solutions. Finally, numerical experiments shows that employing this theoretical guided PINN loss leads to superior accuracy across various benchmarks compared to the standard PINN approach.

Quantitative hydrodynamic limit of the Chern--Simons--Higgs system

Bora Moon Yonsei University Korea Co-Author(s):    Jeongho Kim

Abstract: We study the hydrodynamic limit of the Chern--Simons--Higgs system, a relativistic Chern--Simons gauge field model. We introduce a single scaling parameter capturing both the non-relativistic (infinite speed of light) and semi-classical (vanishing Planck constant) regimes. This unified scaling allows us to justify the simultaneous non-relativistic and semi-classical limit, while retaining the nontrivial effect of the Chern--Simons gauge structure. Using a modulated energy method, we establish quantitative convergence rates toward the corresponding compressible Euler--Chern--Simons system as the scaling parameter tends to zero.

On rotating solutions for Euler-Poisson equations

Jinmyoung Seok Seoul National University Korea Co-Author(s):    Juhi Jang

Abstract: n this talk, I will present some ideas for constructing single and multi-body rotating solutions of Euler-Poisson equations and establishing their nonlinear dynamical stability. 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.

Homogenization of non-divergence type equation with oscillating coefficients defined on a highly oscillating obstacles.

Minha Yoo National Institute for Mathematical Sciences Korea Co-Author(s):    Sunghoon Kim, Ki-Ahm Lee, Se-Chan Lee

Abstract: In this talk, we discuss the homogenization of a highly oscillating obstacle problem using the viscosity method. The equation we deal with is a non-divergence type equation with oscillating coefficients. To analyze the behavior of solutions in the obstacle problem, we construct a corrector function, periodic function when the obstacle is given as 1. By utilizing this corrector, we identify the so-called strange term behavior when the size of the domain where the obstacle is defined reaches a critical value. We then modify the corrector for critical size and analyze the solution`s behavior when the size of the obstacle is either larger or smaller than the critical value.