Junwha Jung, Brown University

Title: Diffusive limit of Boltzmann Equation in exterior Domain

Abstract: The study of flows over an obstacle is one of the fundamental problems in fluids. In this talk we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality in the unbounded domain, we develop a new $L^2-L^6$ splitting to extend the $L^2-L^\infty$ framework into the unbounded domain.

Zoom link:  https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Jaeho Choi, University of Pennsylvania

Title: On the Two-Phase Stokes Flow Problem with Surface Tension 

Abstract: The Navier-Stokes problem with surface tension is a classical free boundary problem in fluid mechanics. After a brief history and overview of this problem, new analytical results on the well-posedness of a quasi-stationary approximation of the Navier-Stokes problem with surface tension will be given. Numerics based on a boundary integral (BI) method will be presented to complement the analytical results.

Min Jun Jo, Duke University

Title: Discussion on the regularity and the singularity in the fluid equations with/without the viscosity effect

Abstract: After briefly reviewing the recent results on the partial regularity for the hyperdissipative Navier-Stokes equations, we discuss the gap between the viscous systems and the non-viscous systems in terms of the issue of regularity/singularity. If time permits, inviscid limit might be treated as an example of the bridge over the spectrum of viscosity.

Dingqun Deng, Jin Woo Jang, and Donghyun Lee

[PDE reading seminar] Weakly collisional regime and Landau damping

The paper we will be studying is 'THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME', published in the JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, available at https://doi.org/10.1090/jams/1014. This topic is very new and interesting as it presents results on Landau damping observed in the Vlasov-Poisson equation when the Landau effect, which causes dissipation, is weak in the Vlasov-Poisson-Landau equations. It's a topic that offers great potential for various extensions in research.

Jaemin Park, University of Basel

Title: Small scale creation for the 2D Boussinesq Equations

Abstract: In this presentation, we will examine the long-term behaviors of the two-dimensional incompressible Boussinesq equations without thermal diffusion. While the Boussinesq equations exhibit energy conservation, the flow's small-scale creation may induce the growth of finer norms in the solutions over time. During this talk, we will construct smooth initial data to demonstrate such norm growth phenomena, both with and without considering kinematic viscosity. Additionally, we will extend our results to provide an example of temperature patch solutions, the curvature and perimeter of which increase as time progresses.  This work is a joint work with A. Kiselev and Y. Yao.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Jiwoong Jang, University of Wisconsin-Madison

Title: Convergence rate of periodic homogenization of forced mean curvature flow of graphs in the laminar setting

Abstract: Mean curvature flow with a forcing term models the motion of a phase boundary through media with defects and heterogeneities. When the environment shows periodic patterns with small oscillations, an averaged behavior is exhibited as we zoom out, and providing mathematical treatment for the behavior has received a great attention recently. In this talk, we discuss the periodic homogenization of forced mean curvature flows, and we give a quantitative analysis for the flow starting from an entire graph in a laminated environment.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Tak Kwong Wong, University of Hong Kong

Title: On the Well-posedness of Classical Solutions to Hamilton-Jacobi-Bellman Equation Arising from the Optimal Savings and the Value of Population Problem under a Stochastic Environment

Abstract: In the work of Arrow et al. (2007, Proc. Natl. Acad. Sci. U.S.A.), they studied a macroeconomic growth model so that the population dynamic was involved in both the total utility (objective function) of the whole population and in the capital investment process. In essence, they assumed the deterministic evolution for both dynamics, such that the labour force of the population is also incurred through the Cobb-Douglas production function. In this talk, we will first introduce an extension of their problem, particularly over a finite time horizon, in which we also allow more realistic and generic population growth and incorporate a stochastic environment for both the demography and capital investment. For the corresponding Hamilton-Jacobi-Bellman equation, we show the existence and uniqueness of classical solutions by using a hybrid approach that combines techniques in both partial differential equations and stochastic analysis. We believe that the methodology developed in this work can also apply to various sophisticated models arising from economic growth theory and mathematical finance. 

This work is supported by the Hong Kong General Research Fund “Controlling the Growth of Classical Solutions of a Class of Parabolic Differential Equations with Singular Coefficients: Resolutions for Some Lasting Problems from Economics” with project number 17302521. 

Reference: Arrow, K., Bensoussan, A., Feng, Q., and Sethi, S.P. (2007). Optimal savings and the value of population, Proceedings of the National Academy of Sciences, 47: 18421-6.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Min Jun Jo, University of British Columbia

Title: Generalized inviscid damping via Fourier analytic control in some 2D fluid models

Abstract: Inviscid damping was named after Landau damping as its hydrodynamic analogue, originally referring to the nonlinear vorticity mixing effect by the shear flows in the 2D Euler equations. In this talk, we broaden the realm of inviscid damping by viewing it as the extra damping mechanism that emerges near certain stationary solutions to the inherently non-dissipative fluid systems. To make it applicable to the low-regularity settings, in contrast to the previous works necessitating high-regularity, we devise a specific kind of Fourier analytic energy estimates to obtain the various types of stability for the target fluid equations. This talk is based on the joint works with Junha Kim and Jihoon Lee.

Kihyun Kim, Institut des hautes études scientifiques  (IHÉS)

Title: Rigidity of long-term dynamics for the self-dual Chern-Simons-Schrödinger equation within equivariance

Abstract: We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. Being a gauged 2D cubic nonlinear Schrödinger equation (NLS), (CSS) is L2-critical and has pseudoconformal invariance and solitons. However, there are two distinguished features of (CSS), the self-duality and non-locality, which make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one(!) modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Yong Wang, Chinese Academy of Sciences

Title: On the global well-posedness and hydrodynamic limit of Boltzmann equation

Abstract: In this talk, we first discuss some results on the global existence and asymptotic behavior for the solutions of Boltzmann equation. Then we will discuss the hydrodynamic limit of Boltzmann equation with specular boundary condition in half-space to the compressible Euler equations.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Min Jun Jo, University of British Columbia

Title: Instantaneous velocity blow-up for the 2D Euler equations in the critical spaces

Abstract: We construct an initial data $u_0\in C^1 \cap H^2$ such that the corresponding velocity field $u$ of the unique Yudovich solution of the 2D Euler equations escapes both $C^1$ and $H^2$ instantaneously. The vorticity-dynamical nature of our proof of $C^1\cap H^2$-illposedness provides a quantitative and non-hypothetical description of the 2D Euler flows. Our study includes the previous illposedness results by Bourgain-Li (2015) and Elgindi-Masmoudi (2020).

Van Tien Nguyen (National Taiwan University)

Title: Singularities in the L^1 critical and supercritical Keller-Segel system

Abstract: In this talk I will present constructive examples of blowup solutions to the Keller-Segel system in R^d. • L^1-critical (d = 2): There exist finite time blowup solutions that are of Type II with finite mass. Blowup rates are quantized according to the spectrum of a linearized operator in the self-similar setting. There is also the case of multiple collapsing blowup solutions formed by a collision of single-solutions. • L^1-supercritical (d ≥ 3): We exhibit finite time blowup solutions that are completely unrelated to the self-similar scale, in particular, they are of Type II with finite mass. Interestingly, the radial blowup profile is linked to the traveling wave of the 1D viscous Burgers equation. There also exist solutions that blow up in finite time with infinite mass. The solution is asymptotically self-similar with a logarithmic correction to its profile for d = 3, 4. We found such an asymptotic profile can be either radial or completely non-radial. The talk is based on results obtained in collaboration with Collot (Paris Cergy), Ghoul (NYU Abu Dhabi), Masmousdi (NYU), Nouaili (Paris Dauphine), Zaag (Paris Nord). 

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Satoshi Taguchi, Kyoto University

Title: A Stokes system with a source/sink-type condition derived from a Boltzmann system with discontinuous boundary data

Abstract: In this talk, we consider a boundary-value problem of the Boltzmann equation with discontinuous boundary data and its fluid-dynamic approximation in the near continuum regime. More precisely, we consider the steady gas flow between two parallel walls induced by a discontinuous wall temperature. Our motivation for this study is a recent interest in phoretically active particles, typically known as Janus particles. Assuming a small Knudsen number, we derive from the Boltzmann equation a system of Stokes equations for incompressible fluids with a source/sink condition to model the effect of the discontinuous wall temperature. The derivation is based on a matched asymptotic expansion method, which allows us to identify a kinetic region near the point of discontinuity. The present talk is based on a joint work with Tetsuro Tsuji.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Changkeun Oh, MIT

Title: Decoupling inequalities for quadratic forms

Abstract: Decoupling inequalities are known to be powerful tools in the study of the solution counting of a system of Diophantine equations. In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Sangdon Jin, Chung-Ang University

Title: On steady states for the Vlasov-Schrödinger-Poisson system

Abstract: The Vlasov-Schrödinger-Poisson system is a kinetic-quantum hybrid model describing quasi-lower dimensional electron gases. In this talk, we discuss the derivation of kinetic quantum hybrid models using partial confinement. Also, for this system, we study the construction of a large class of 2D kinetic/1D quantum steady states in a bounded domain as generalized free energy minimizers, and we show their finite subband structure, monotonicity, uniqueness, and conditional dynamical stability. This talk is based on joint work with Younghun Hong  (Chung-Ang University).

Jacky Chong, Peking University

Title: Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons

Abstract: In this talk, we study the effective (mean-field) dynamics of a many- body bosonic system and obtain a quantitative bound on the error between the exact and effective dynamics. More precisely, we consider a system of N interacting bosons where the particles experience a short range two-body interaction given by N −1 vN (x) = N3β−1 v(Nβx), which scales to a δ function, where 0 < β < 1 and v is a non-negative spherically symmetric function. Our main result is the extension of the local-in-time Fock space norm approximation of the exact dynamics of quasifree states proven in M. Grillakis and M. Machedon, Comm. PDEs., 42, 24(2017) for 0 < β < 2 3 to a global-in-time approximation for 0 < β < 1. Our extension allows for a more general set of initial data that includes coherent states. The key ingredients in establishing the Fock space approximation are the work of Grillakis and Machedon on the local well-posedness theory for the time-dependent Hartree–Fock– Bogoliubov (HFB) system in M. Grillakis and M. Machedon, Comm. PDEs., 44, 1431(2019) and a global-in-time estimate for the HBF system in J. Chong, M. Grillakis, M. Machedon, and Z. Zhao, Comm. PDEs., 46, 2015(2021). This is a joint work with Z. Zhao, Ann. Henri Poincaré (2021). The talk will be delivered in English.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Haitao Wang, Shanghai Jiao Tong University

Title: Propagation of rough initial data for Navier-Stokes equation

Abstract: In this talk, I will present a quantitative study of a weak solution for an initial value problem of the compressible Navier-Stokes equation in the class of BV functions. The key tool in the proof is the ``effective Green's function'', which is an interpolation between heat kernel for BV coefficient and Green's function for linearized Navier-Stokes equation.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Ayman R. Said, Duke University

Title: On the long-time behavior of scale-invariant solutions to the 2d Euler equation

Abstract: In these 2 lectures we will give a brief introduction to the 2d Euler equations and discuss some of the key open questions in the field today. Then we will present our recent findings, in which we will give a complete description of the long-time behavior of uniformly bounded regulated scale-invariant solutions to the 2d Euler equation satisfying a discrete symmetry. We show that all such solutions relax in infinite time to rigidly rotating or steady states, which are fully classified and shown to be piece-wise constant profiles with countably many jumps. Consequently, all sufficiently symmetric non-constant scale-invariant solutions that are smooth on S 1 become singular in infinite time. On R 2, this corresponds to a generic infinite time spiral and cusp formation. In the process, we also show that for scale-invariant solutions, the measure (on S 1 ) of particles moving away from the origin and toward spatial infinity is a strictly increasing function of time. This increasing function of time generalizes to general solutions of the 2d Euler equation that are bounded and m-fold symmetric (m ≥ 4).

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Junha Kim, Korea Institute for Advanced Study (KIAS)

Title: Asymptotic stability near hydrostatic equilibrium to the Boussinesq equations

Abstract: The Boussinesq equations are frequently used to study the dynamics of the ocean or the atmosphere. In this talk, we consider Boussinesq equations without thermal diffusion in the domain $\bbT^{d-1} \times [-1,1]$, $d \geq 2$. We show that the stationary stratified solution $(v_s,\rho_s,p_s) = (0,x_d,x_d^2/2)$ is stable. Then, we provide sharp temporal decay estimates of the solution. This is joint work with Prof. Juhi Jang (USC).

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Renjun Duan, The Chinese University of Hong Kong

Title: Low-regularity solutions to the non-cutoff Boltzmann equation in $\mathbb{R}^3$

Abstract: A class of low-regularity solutions via the Wiener algebra for the non-cutoff Boltzmann equation in torus was previously introduced in collaboration with Liu, Sakamoto and Strain. In the talk, I will report a recent work, joint with Sakamoto and Ueda, for how to extend the result to the case of the whole space. In this case, we develop an $L^1 \cap L^p$ interplay approach in the Fourier space to overcome the weaker macroscopic dissipation due to diffusion phenomenon in contrast to the torus case. The key is to employ time-weighted estimates motivated from the study of viscous conservation laws.

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Seunghyeok Kim, Hanyang University

Title: Regularity theory of local and nonlocal elliptic equations 

Abstract: In this two-day lecture series, we will be concerned with the regularity theory of second-order elliptic equations and certain nonlocal elliptic equations. On the first day, we will recall the definition of second-order elliptic equations, review the history of classical regularity theory involving them, and sketch the proof of Hölder regularity for their weak solutions. On the second day, we will introduce several nonlocal elliptic equations, discuss why such equations have become popular in the research field, and sketch the proof of Hölder regularity for weak solutions of basic nonlocal elliptic equations. 

Lecture 1. A brief history of regularity theory for elliptic equations (Schauder estimates, De Giorgi-Nash-Moser estimates, L^p theory) 

Lecture 2. Hölder regularity for weak solutions of second-order elliptic equations 

Lecture 3. Nonlocal elliptic equations 

Lecture 4. Hölder regularity for weak solutions of nonlocal elliptic equations

Halyun Jeong, UCLA

Title : Concentration of sub-Gaussian matrices on sets: optimal tail decay and applications. 

Abstract : Random linear mappings are ubiquitous in dimension reduction, compressed sensing, group testing, and numerical linear algebra. Due to the computational benefits of non-Gaussian mappings, their performance has received considerable attention from researchers. This can be captured by the worst-case deviation error from an isometry and for isotropic sub-Gaussian random matrices, the error is given in terms of the sub-Gaussian norm $K$. The previous best-known dependency of the error on the sub-Gaussian norm is $K^2$ but it was not known that it is optimal. In joint work with Xiaowei Li, Yaniv Plan, and {\"O}zg{\"u}r Y{\i}lmaz, using our new Bernstein's inequality and Hanson-Wright inequality, we were able to improve the error dependency $K^2$ to $K \sqrt{ \log K}$ and proved that it also is optimal. Consequently, our result offers better theoretical guarantees for several applications such as Johnson-Lindenstrauss embedding, random sketches for numerical linear algebra, and sparse recovery from $0/1$ matrices. Our new Bernstein's inequality has been also applied to lower the sample complexity of the blind demodulation with deep generative neural network priors. In this talk, I will present the idea of our work and discuss its applications.

Chanwoo Kim, University of Wisconsin-Madison

Title: Boundary effect on Stability 

Abstract: In this lecture, we present some recent asymptotic stability results of Vlasov equations induced by the boundary.

Quoc-Hung Nguyen, Chinese Academy of Sciences (CAS)

Title: Nonlinear Landau damping for the 2d/3d Vlasov-Poisson system with massless electrons around Penrose-stable equilibria

Abstract: In this talk, we discuss the asymptotic stability of Penrose-stable equilibria among solutions of the Vlasov-Poisson system with electrons mass in $\mathbb{R}^d$ with $d\geq 2.$

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Hyunwoo Kwon, Sogang University

Title: Survey on Landau damping

Abstract: In this survey talk, we consider one of the interesting properties of the solution of the Vlasov-Poisson equation, the Landau damping. Since the prediction of Landau (1946), many physicists and mathematicians have devoted themselves to manifesting this mysterious phenomenon and proving this result in rigorous settings. Mouhot-Villani (2011) first rigorously established this phenomenon in analytic settings. Later, it was extended by Bedrossian-Masmoudi-Mouhot (2016) in Gevrey settings. In this survey talk, we follow the work of Bedrossian-Masmoudi-Mouhot. We provide necessary preliminary materials to understand the result.

Tak Kwong Wong, University of Hong Kong

Title: Training Deep ResNet with Batch Normalization as a First-Order Mean-Field Type Control Problem

Abstract: Inspired by the architecture of human and animal brains, artificial neural networks (ANN) were introduced, and have been extensively studied since 1940s. Due to a tremendous amount of real-world applications, the development of this area is extremely fast in recent decades. In order to avoid technical problems, such as vanishing gradients, residual neural networks (ResNet) with batch normalization are commonly used in various practical applications. On the other hand, as a generalization of calculus of variations, the optimal control theory aims at finding a control for a given dynamical system, so that an objective function is optimized. When the number of agents is large, the mean-field approximation applies and this will lead to the mean-field type control problems, which have many applications in economics and engineering and are extremely challenging problems in the mathematical analysis. In this talk, we will first explain the connection between ResNet with batch normalization and the first-order mean-field type control problems, and then discuss our new results about solving the first-order mean-field type control problems in a generic setting. This is a joint work with Alain Bensoussan, Phillip Yam, and Hongwei Yuan.

This work is supported by Hong Kong General Research Fund (GRF) grant “Solving Generic Mean Field Type Problems: Interplay between Partial Differential Equations and Stochastic Analysis” with project number 17306420. 

Zoom link: https://postech-ac-kr.zoom.us/j/96428451691?pwd=YTdPQTMrd1JlVUlzUkk0ZUZkRVM2dz09

Bongsuk Kwon, UNIST

Title: Singularity formation in plasma ion dynamics

Abstract: We consider the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for the singularity formation of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the presureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can break down even if the gradient of initial velocity is trivial. For the isothermal case, we prove that smooth solutions break down in a finite time when the gradients of the Riemann invariants are initially large. If time permits, we will discuss the blow-up profiles for the Burgers equation and their properties, which will be a key step toward further investigation of singularity formation for the Euler-Poisson system. This is joint work with J. Bae (NCTS, Taipei) and J. Choi (SKKU).

Jinmyoung Seok, Kyonggi University

Title: Connections of special solutions to nonlinear PDEs in the limit processes from math physics

Abstract: In this talk, I first introduce the limit processes arising from mathematical physics such as the mean field limit, the nonrelativistic limit, the semi-classical limit and the hydrodynamic limit, which connect different physical regimes. In mathematical theory of nonlinear PDEs, these limit processes give a beautiful picture connecting various fundamental nonlinear PDEs. The rigorous derivations of these PDEs through the limit processes have been one of fundamental and important research topics in mathematical physics and PDE communities. This talk especially focuses on the connections between variationally constructed special solutions to nonlinear PDEs in the processes of limits. These special solutions describe interesting physical phenomena such as the Bose-Einstein condensation or the ionization problem in the atomic level and the formation of white dwarfs, binary stars and even hypothetical boson stars in astrophysical level. The aim of this talk is to explain, in a limited range based on my previous works, how these connections between special solutions can affect and answer the important questions in PDE theories such as the existence and uniqueness and stability of them.

Young-Pil Choi, Yonsei University

Title: Well-posedness and singularity formation for Vlasov-Riesz system

Abstract: In this talk, we discuss the Cauchy problem for the Vlasov-Riesz system, which is a Vlasov equation featuring interaction potentials generalizing various previously studied cases, including the Coulomb and Manev potentials. For the first time, we extend the local theory of classical solutions to interaction potentials which are more singular than that for the Manev. Then, we obtain finite-time singularity formation for solutions with various attractive interaction potentials, extending the well-known singularity formation result for attractive Vlasov-Poisson. Our local well-posedness and singularity formation results extend to cases with linear diffusion and damping in velocity.

Zoom link: https://postech-ac-kr.zoom.us/j/6629787929?pwd=NVFKemFpOE5kZlpld0E5MldjaHk2Zz09