Program

I will present recent works concerning the stability of a Riemann weak shock solution to the isothermal Euler system within the framework of inviscid limits from the Navier-Stokes system. This study extends previous work conducted for the isentropic case to encompass the isothermal scenario.

We investigate the formation of singularities in the regularized Saint-Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of these singularities has not been rigorously characterized. In this work, we establish the stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics near the singularity and proves sharp $C^{3/5}$ H\"older regularity at blow-up. This regularity differs from the $C^{1/3}$ cubic-root singularities found in the compressible Euler and inviscid Burgers equations. The difference in blow-up regularity between $C^{1/3}$ in the Euler equations and $C^{3/5}$ in the rSV system highlights the structural influence of Hamiltonian regularization on singularity formation. This is joint with Yunjoo Kim at UNIST and Wanyong Shim at KAIST.

In recent years, modern machine learning techniques using deep neural networks have achieved tremendous success in various fields. From a mathematical point of view, deep learning essentially involves approximating a target function, relying on the approximation power of deep neural networks. Therefore, it is important to understand the approximation and generalization properties of neural networks in high dimensions. The primary objective of this talk is to mathematically analyze the approximation of neural networks within the classical numerical analysis framework. We will explore the proper regularity of target functions which is suitable for neural network approximation, and investigate how these properties are reflected in the approximation and learning complexity of neural networks. Next, I will apply these theories to my recent work on the operator learning method for solving parametric PDEs. I will analyze the intrinsic structure of the proposed method through the theory described above, deriving some useful results both theoretically and practically. Furthermore, I will demonstrate some relevant numerical experiments, confirming that these theory-guided strategies can be utilized to significantly improve the performance of the method.

In this talk, we discuss the rigorous derivation of isentropic two-phase fluid equations from the kinetic model for mixtures. For this, we propose a BGK-type kinetic model for a binary gas mixture by solving the entropy minimization problems. Then we perform a formal Chapman-Enskog expansion under a hydrodynamic scaling, which serves to identify the macroscopic structure associated with the model. To complement the formal analysis, we carry out a rigorous derivation of the isentropic two-phase Euler equations using the relative entropy method. To the best of our knowledge, a number of works have addressed the rigorous derivation of macroscopic two-phase equations starting from kinetic-fluid systems, but a complete rigorous justification from kinetic mixtures has remained open. Our result will provide the first such derivation of the isentropic two-phase flows from kinetic mixtures, making precise the connection between mesoscopic relaxation mechanisms and macroscopic fluid dynamics. This is joint work with S. Y. Cho, Y.-P. Choi, and S. Song.

In this talk, we will discuss Leray-Hopf solutions to the incompressible Navier-Stokes equations with vanishing viscosity. We explore important features of turbulence, focusing around the anomalous energy dissipation phenomenon. As a related result, I will present a recent result proving that for two-dimensional fluids, assuming that  the initial vorticity is merely a Radon measure with nonnegative singular part, there is no anomalous energy dissipation. Our proof draws on several key observations from the work of J. Delort (1991) on constructing global weak solutions to the Euler equation. We will also discuss possible extensions to the viscous SQG equation in the context of Hamiltonian conservation and existence of weak solutions for a rough initial data. This is a joint work with MiKael Latocca (Univ. Evry) and Luigi De Rosa (GSSI).

In this talk, we investigate the Cauchy problem of a certain type of non-Newtonian inhomogenous Navier-Stokes equations in three dimensions. We establish local existence of unique regular solutions for sufficiently smooth initial data under the compatible condition via the energy estimate. This is a joint work with Kyungkeun Kang and Hwa Kil Kim.

One of the central questions in collisionless plasma physics is analyzing the long-time behavior of solutions to the Vlasov equations. Specifically, constructing space-inhomogeneous equilibria and establishing their stability is a major, long-standing open challenge, particularly when magnetic fields are involved. Motivated by solar wind models, we will study a long-time behavior of the three-dimensional Vlasov-Maxwell system under an ambient gravitational field, for two-species particle density distributions. We prove the uniqueness of the steady solution within a regularity space to which we can rigorously show the solution belongs. We then construct a global-in-time solution near these equilibria and establish their asymptotic stability under general small perturbations. This is the first result on asymptotic stability for non-vacuum equilibria in the full Vlasov-Maxwell system in 3D. This is a joint-work with Chanwoo Kim.

The conservation of energy is one of the fundamental laws of physics. While the motion of a physical object may increase or decrease a particular form of energy, the total energy remains constant. For example, when a fluid flows downhill, gravitational potential energy is converted into an equal amount of kinetic energy. This transfer between gravitational potential and kinetic energy is well known and familiar even to non-scientists. In contrast, the transfer between surface energy and kinetic energy, when surface tension plays a significant role, is much less widely recognized. This talk presents several numerical methods for solving the incompressible Navier–Stokes equations with a certain level of stability guarantee, covering applications in multiphase flow, surface tension, and fluid–solid interaction.

In this talk, we are interested in uniformly rotating compressible fluid bodies coupled with an external particle, which can be regarded as a model for a star–planet system. This is described by a rotating solution of coupled system (EPN) consisting of the gravitational Euler–Poisson equations for the fluid body and Newton’s equation for the particle. We show that, for any prescribed positive number d>0, there exists a stationary solution of (EPN) in a uniformly rotating frame such that the distance between the centers of mass of the fluid and the particle is exactly d. We will then discuss aspects of the nonlinear stability of these solutions.

In this talk, we present a quantified analysis of the hydrodynamic limit for a relativistic quantum system interacting with self-consistent electromagnetic fields, specifically the Maxwell–Klein–Gordon (MKG) system. We first establish global well-posedness of the MKG system under the Coulomb gauge, even in the presence of a self-interaction potential. To derive the hydrodynamic limit, we introduce a single scaling parameter that simultaneously captures both the non-relativistic and semi-classical regimes by controlling the speed of light and Planck’s constant. As this parameter tends to zero, we rigorously obtain quantified convergence estimates from the electrostatic MKG system to the classical Euler–Poisson system, using the modulated energy method.