Special Session 76: Recent Developments in Nonlinear and Nonlocal Evolution Equations

The relativistic quantum Boltzmann equation near equilibrium

Gi-Chan Bae Seoul National University / Research institute of Mathematics Korea Co-Author(s):    Jin Woo Jang, Seok-Bae Yun

Abstract: The relativistic quantum Boltzmann equation (or the relativistic Uehling-Uhlenbeck equation) describes the dynamics of single-species fast-moving quantum particles. With the recent development of the relativistic quantum mechanics, the relativistic quantum Boltzmann equation has been widely used in physics and engineering such as in the quantum collision experiments and the simulations of electrons in graphene. In spite of such importance, there has been no mathematical theory on the existence of solutions for the relativistic quantum Boltzmann equation to the best of authors` knowledge. In this talk, we consider the global existence of a unique classical solution to the relativistic Boltzmann equation for both bosons and fermions when the initial distribution is nearby a global equilibrium. This is joint work with J. W. Jang and S. B. Yun.

Dynamics and Convergence Arising from Some Phase Field Models

Yuan Chen Chinese University of Hong Kong Peoples Rep of China Co-Author(s):

Abstract: We consider the mass-preserving $L^2$ -gradient flow of the weak or strong scaling of the functionalized Cahn-Hilliard equation and justify its sharp interface limit. With a suitable mass condition, the accumulated material forms a bilayer interface with width $\varepsilon$, which balances with the bulk phase. In the weak scaling case, we rigorously demonstrate that for well-prepared initial data, as the interface width $\varepsilon$ tends to zero, the bilayer interface converges to an area-preserving Willmore flow. This result holds for any dimension $n$ .

On the dynamics of surface waves for a fluid with odd viscosity

Rafael Granero Belinchon Universidad de Cantabria Spain Co-Author(s):    Diego Alonso-Oran, Claudia Garcia, Alejandro Ortega

Abstract: In this talk we will review some recent results on the dynamics of a free boundary problem arising in a non-newtonian viscous flow with the so called odd viscosity. This viscosity is also known as Hall viscosity and appears in a number of applications such as quantum Hall fluids or chiral active fluids. Besides the odd viscosity effects, the models that we will present capture both gravity and capillary forces up to quadratic interactions and take the form of nonlinear and nonlocal wave equations. We will present some well-posedness result and also study the existence of m-fold symmetric traveling waves.

Singularity formation and global weak solutions to the Serre-Green-Naghdi equations with surface tension

Billel Guelmame ENS Lyon France Co-Author(s):

Abstract: In this talk, we explore the Serre-Green-Naghdi equations, which describe shallow-water waves while considering the influence of surface tension. These equations are locally (in time) well-posed. We identify a class of smooth initial data, leading to the development of singularities in finite time for the corresponding strong solutions. Additionally, we demonstrate the existence of global weak solutions for small-energy initial data.

Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

SOYEUN JUNG Kongju National University Korea Co-Author(s):    Pavel Dr\`{a}bek, Eunkyung Ko, Michaela Zahradn\`{i}kov\`{a}

Abstract: In this talk we consider wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection.

Some results on a repulsive chemotaxis-consumption model

Dongkwang Kim Ulsan National Institute of Science and Technology, Department of Mathematical Sciences Korea Co-Author(s):    Jaewook Ahn, Kyungkeun Kang

Abstract: In this talk, we will discuss results concerning the solvability of a chemotaxis model, which describes the movement of organisms in response to chemical substances. Focusing on the repulsive chemotaxis-consumption system, we examine the criteria under which solutions remain bounded over time and the conditions leading to blow-up in higher dimensions. Specifically, we show that the system admits globally bounded solutions when the diffusion of the organisms is enhanced, or when the diffusion is weakened but the boundary data for the signal substance is sufficiently small. On the other hand, we prove that if the diffusion is further weakened and the boundary data for the signal is sufficiently large, the system exhibits blow-up behavior.

Blowup solutions to the complex Ginzburg-Landau equation

Van Tien Nguyen National Taiwan University Taiwan Co-Author(s):    Jiajie Chen, Thomas Y. Hou, Yixuan Wang

Abstract: We develop a so-called generalized dynamical rescaling method to study singularity formation in the complex Ginzburg-Landau equation (CGL). This innovative technique enables us to capture all relevant symmetries of the problem, allowing us to directly demonstrate a full stability of constructed blowup solutions. One of the advantages of our approach is its ability to circumvent spectral decomposition, which is often complex for problems involving non-self- adjoint operators. Additionally, the (CGL) system lacks a variational structure, making standard energy-type methods difficult to apply. By employing the amplitude-phase representation, we establish a robust analysis framework that enforces vanishing conditions through a carefully chosen normalization and utilizes weighted energy estimates.

Liouville-type theorems for the stationary ideal magnetohydrodynamics equations in multi-dimensional cases

Anthony Suen The Education University of Hong Kong Hong Kong Co-Author(s):    Lv Cai, Ning-An Lai, Manwai Yuen

Abstract: We establish Liouville-type theorems for the stationary ideal compressible magnetohydrodynamics system in $\mathbb{R}^n$ with $n\in\{1, 2, 3\}$. We address various cases when the finite energy condition is in force and the stationary density function $\rho$ satisfies $\lim_{|x|\to\infty}\rho(x)=\rho_\infty\ge0$. Our proof relies heavily on the good structure of the nonlinear magnetic force term and the usage of well-chosen smooth cut-off test functions.

Mean Field Games, FBSDEs and Associated Master Equations

Ho Man Tai Dublin City University Ireland Co-Author(s):    Alain Bensoussan, Ho Man Tai, Tak Kwong Wong, and Sheung Chi Phillip Yam

Abstract: In this talk, I introduce the global-in-time well-posedness for a broad class of mean field game problems, which is beyond the special linear-quadratic setting, as long as the mean field effect is not too large. Through the stochastic maximum principle, we adopt the forward backward stochastic differential equation (FBSDE) approach to investigate the unique existence of the corresponding equilibrium strategies. Further analysis on the Jacobian flow of the FBSDE will be discussed so as to establish the classical well-posedness of the master equation on $R^d$.

3D hard sphere Boltzmann equation: explicit structure and the transition process from polynomial tail to Gaussian tail

Haitao Wang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Yu-Chu Lin, Kung-Chien Wu

Abstract: We study the Boltzmann equation with hard sphere in a near-equilibrium setting. The initial data is compactly supported in the space variable and has a polynomial tail in the microscopic velocity. We show that the solution can be decomposed into a particle-like part (polynomial tail) and a fluid-like part (Gaussian tail). The particle-like part decays exponentially in both space and time, while the fluid-like part dominates the long time behavior and exhibits rich wave motion. The nonlinear wave interactions in the fluid-like part are precisely characterized. Furthermore, the transition process from the polynomial to the Gaussian tail is quantitatively revealed.