Special Session 80: Nonlinear dynamics of particle systems and fluids

High Reynolds number limit of 2D Boltzmann equation

Gi-Chan Bae Seoul National University / Research institute of Mathematics Korea Co-Author(s):    Chanwoo Kim

Abstract: We prove the hydrodynamic limit of the 2D Boltzmann equation to the incompressible Euler equation in a periodic box. This is joint work with Chanwoo Kim.

Mathematical Analysis of Some Models of Active Matter

Hantaek Bae Ulsan National Institute of Science and Technology Korea Co-Author(s):    Young-Pil Choi, Kyungkeun Kang, Woojae Lee

Abstract: Active matter theories have made remarkable progress in understanding the dynamics of suspension of active polar particles such as fish school, locust swarm, and bird flock. The large scale behavior of active matter systems can be described by continuum models which describe the evolution of slow variables such as the number density and the velocity field. In this talk, we introduce some some recents results on these continuum models.

Interaction of Rigid Ball and Incompressible Fluid

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

Abstract: We talk the large time behavior of the solutions of the Stokes fluid-solid system. We compute the asymptotic expansion of the solution for $L^q$ integrable initial data $U_0$. We show that the asymptotic profile of the solution is a linear combinations of the Stokes fundamental solution for the data with extra condition $|x|U_0\in L^1$. We also show that $L^1$ integrability of the solution is strongly related to the net force exerted by the fluid on the boundary of the solid.\

High order conservative semi-Lagrangian schemes for the ES-BGK model of the Boltzmann equation

Seung Yeon Cho Department of Mathematics, Gyeongsang National University Korea Co-Author(s):    Sebastiano Boscarino, Giovanni Russo, Seok-Bae Yun

Abstract: In this talk, we introduce finite difference high order conservative semi-Lagrangian schemes for the ellipsoidal BGK model of the Boltzmann equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator with small Knudsen number, we employ high order $L$-stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit scheme is designed to update solutions explicitly without resorting to any Newton solver. We present several numerical tests to demonstrate the accuracy and efficiency of the proposed method. In particular, we show that our method is able to capture the behavior of Navier-Stokes equations for moderate values of Knudsen number, and provide good approximation of the solution to Boltzmann equation for relatively large values of Knudsen number.

Liouville-type theorems for the stationary Navier-Stokes equations

Youseung Cho Yonsei University Korea Co-Author(s):    Jiri Neustupa, Minsuk Yang

Abstract: In this talk, we consider the Liouville-type theorems for the stationary Navier-Stokes equations. The classical Liouville theorem states that any bounded and entire holomorphic function must be constant. In recent times, the Liouville theorem has been developed for elliptic equations. We discuss the conditions that guarantees the triviality of the solution of the Navier-Stokes equations. This talk is based on the joint work with Professor Ji\v{r}\`{\i} Neustupa and Minsuk Yang.

Emergent dynamics of infinitely many Kuramoto oscillators

Seung Yeal Ha Seoul National University Korea Co-Author(s):    Euntaek Lee, Woojoo Shim

Abstract: In this talk, we propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender`s index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University).

Local bifurcation for the one dimensional Gray-Scott model

Jongmin Han Kyung Hee University Korea Co-Author(s):    Yuncherl Choi, Taeyoung Ha, Sewoong Kim, Doo Seok Lee

Abstract: In this talk, we consider the local bifurcation near a ciritical control parmanter for the one dimensional Gray-Scott model. We study the bifurcation at both simple and double eigenvalues of the linearized operator at a constant solution. We show that the constant solution bifurcates to an attractor which determines final patterns of solutions.

Vanishing angular singularity limit for the Boltzmann equation without angular cutoff

Jin Woo Jang POSTECH Korea Co-Author(s):    Bernhard Kepka, Alessia Nota, Juan J. L. Velazquez

Abstract: In this talk, we will discuss the vanishing angular singularity limit of the Boltzmann equation. We first recall the derivation of Boltzmann`s collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $d=3$. Then we study the limit of the non-cutoff kernel to the hard-sphere kernel. We also give precise asymptotic formulas of the singular layer near the angular singularity in the limit $s\to \infty$. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions weakly in $L^1$ globally in time as $s\to \infty$ by looking at Arkeryd`s construction of a weak solution to the Boltzmann equation for hard-sphere collisions and Villani`s construction of an entropy solution for the Boltzmann equation for long-range inverse-power law potential. The spatially inhomogeneous case is still open.

Analysis of score-based diffusion models with multiplicative noise conditioning

Doheon Kim Hanyang University Korea Co-Author(s):    Doheon Kim

Abstract: Score-based diffusion models generate new samples by learning the score associated with a diffusion process. When the score is accurately approximated, the effectiveness of these models can be theoretically justified using differential equations related to the sampling process. Despite this, empirical evidence shows that models employing neural networks with multiplicative noise conditioning can still produce high-quality samples, even when their capacity is clearly insufficient to learn the correct score. We offer a theoretical explanation for this phenomenon by examining the qualitative behavior of the differential equations governing the diffusion processes, utilizing appropriate Lyapunov functions for analysis.

Asymptotic convergence of the heterogeneous first-order aggregation models: from the sphere to the unitary group

Dohyun Kim Sungkyunkwan University Korea Co-Author(s):    Hansol Park

Abstract: In this talk, we establish the convergence toward equilibrium for heterogeneous multi-agent systems on the unit sphere and the unitary group that can be understood as (small) perturbation of gradient flows. Due to the heterogeneity, one could expect that all relative distances converge to definite values and furthermore that each agent converges to a possibly different stationary point. For the desired convergence, we use the lifting method and dimension reduction method for the cases of the unit sphere and unitary group, respectively. This talk is based on the joint work with Dr. Hansol Park (Dalhousie University).

Interpolation inequalities in Lorentz spaces and their applications to a Stokes-Magneto system with fractional diffusions

Hyunseok Kim Sogang University Korea Co-Author(s):

Abstract: It has been recently shown that the classical interpolation inequalities due to Ladyzhenskaya and Gagliardo-Nirenberg can be refined by using weak $L^p$-norms. The goal of the talk is to present further refinements via general Lorentz spaces. We provide interpolation inequalities in Sobolev-Lorentz spaces of arbitrary orders, as special cases of more general results on Triebel-Lizorkin-Lorentz spaces. Then as an application, we study global weak solutions to a Stokes-Magneto system with fractional diffusions.

Asymptotic behavior toward viscous shock for impermeable wall and inflow problems of barotropic Navier-Stokes equations

Jeongho Kim Kyung Hee University Korea Co-Author(s):    Xushan Huang, Moon-Jin Kang, Jeongho Kim and Hobin Lee

Abstract: We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. We do not impose the zero-mass condition on initial data, which improves the previous results by Matsumura and Mei \cite{MM99} for impermeable case, and by Huang, Matsumura and Shi \cite{HMS03} for inflow case. Moreover, for the inflow case, we remove the assumption $\gamma\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems.

Nontopological bubbling solutions for Chern-Simons system of rank 2

Namkwon Kim Chosun university Korea Co-Author(s):

Abstract: We consider Chern-Simons gauge theory of rank two in the whole space. There are three types of solutions for the system. We deal with nontopological solutions among others. To understand the structure of the solutions space, It is helpful to understand bubbling solutions. We present complete classification of bubbling solutions for the system and, in particular, present existence theory for the $SU(3)$, $SO(5)$, and $G_2$ system.

Physics-informed Neural Networks for the Pseudo two dimensional model of Lithium ion battery

Myeong-Su Lee Korea Advanced Institute of Science and Technology Korea Co-Author(s):    Youngjoon Hong, and Jaemin Oh

Abstract: The pseudo-two-dimensional (P2D) model is a mathematical model that describes the electrochemical processes in Li-ion batteries. Thi model is composed of multiple nonlinear partial differential equations and nonlinear relations, such as the Butler-Volmer equation. In this talk, we investegate the application of Physics-informed neural networks(PINNs) for solving P2D model. Due to the aformentioned nonlinearities in the P2D model, the standard approach often lead to inaccurate solutions. To address these issues, we introduce additional strategies: (1) the incorporation of bypassing terms and (2) the implementation of secondary conservation laws, aimed at improving the stability and accuracy of the PINNs. We first show the efficiency and importance of these strategies through simple toy examples. And then, we present simulation results for the P2D model using PINNs enhanced with our proposed strategies.

Stability and optimal temporal decay result for the 3D Boussinesq equations with horizontal dissipation in anisotropic Sobolev spaces

Bataa Lkhagvasuren Chonnam National University Korea Co-Author(s):    Hyeong-Ohk Bae, Bataa Lkhagvasuren

Abstract: In this talk, we consider 3D anisotropic Boussinesq equations with horizontal dissipation. We prove that, for the perturbed equations, the time global solution exists for small initial data in the anisotropic Sobolev spaces $H^{0,s}$ with $\frac 12 < s$ and the corresponding solution of the unperturbed equations approaches the hydrostatic equilibrium. Moreover, the optimal decay result is obtained in the anisotropic Sobolev spaces $H^{0,s}$, extending the result of isotropic Sobolev spaces.