Abstract : Hyperbolic balance laws are of great interest owing to their importance in applications, such as the Euler equations, magnetohydrodynamic system, Boltzmann equations, and in applied mathematics. Recently, there have been a lot of advances in ascertaining the global well-posedness of such systems. The generic breakdown of classical solutions requires one to enlarge the space to hope for well-posedness or add `good' source terms.
The mathematical studies of these PDEs pose significant analytical/numerical challenges. This minisymposium seeks to bring together researchers to promote exchange of ideas, and present recent developments on the mathematical analysis and novel methods in this area.
Organizer(s) : Manas Bhatnagar, Geng Chen, Hailiang Liu
[03611] Existence and Stability of Traveling Waves of Boussinesq-Burgers Equations
Format : Online Talk on Zoom
Author(s) :
Kyle Kun Zhao (Tulane University)
Anita Yang (Chinese University of Hong Kong)
Zhian Wang (Hong Kong Polytechnic University)
Abstract : We introduce rigorous mathematical results concerning the existence and stability of traveling wave solutions to the Cauchy problem of the one-dimensional Boussinesq-Burgers equations modeling the propagation of weak tidal bores. Existence of traveling waves is obtained by means of phase plane analysis and geometric singular perturbation. Local stability of traveling waves with arbitrary strength is proven by spatially weighted energy methods.
[04963] Global dynamics and photon loss in the Kompaneets equation
Format : Talk at Waseda University
Author(s) :
Hailiang Liu (Iowa State University )
Abstract : The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We present several results concerning the long-time convergence of solutions to Bose–Einstein equilibria and the failure of photon conservation due to shock formation at the zero-energy boundary. This talk is based on a joint work with J. Ballew, G. Iyer, D. Levermore and R. Pego.
[03629] HYPOCOERCIVITY OF STOCHASTIC GALERKIN FORMULATIONS FOR STABILIZATION OF KINETIC EQUATIONS
Format : Talk at Waseda University
Author(s) :
Hui Yu (Tsinghua University)
Stephan Gerster (Universit`a degli Studi dell’Insubria)
Michael Herty (RWTH Aachen University)
Abstract : We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the gamma-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.
[04154] Traveling Wave Solutions in Keller-Segel Models of Chemotaxis
Format : Online Talk on Zoom
Author(s) :
Tong Li (The University of Iowa)
Abstract : We study global existence and long-time behavior of solutions for hyperbolic-parabolic PDE models of chemotaxis.
We show the existence and the stability of traveling wave solutions to systems of nonlinear conservation laws derived from the Keller-Segel model. We construct biologically relevant oscillatory traveling wave solutions to an attractive chemotaxis system of mixed-type. Traveling wave solutions of chemotaxis models with growth are also investigated.
[04448] On the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing
Format : Online Talk on Zoom
Author(s) :
Yongki Lee (Georgia Southern University)
Abstract : The multi-dimensional Euler-Poisson system describes the dynamic behavior of many important physical flows. In this talk, a Riccati system that governs pressureless two-dimensional EP equations is discussed. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behavior of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. Several recent works in a similar vein will be reviewed.
[04918] Critical thresholds in spherically symmetric Euler-Poisson systems
Format : Talk at Waseda University
Author(s) :
Manas Bhatnagar (University of Massachusetts Amherst)
Abstract : We will see an introduction to the concept of Critical Threshold Phenomena (CTP) and how it plays a role in the Euler-Poisson systems. We will go over some of the existing results in this area and how the techniques have developed over time. In the end, we will see some new results on the multidimensional spherically symmetric Euler-Poisson systems.
[04518] On multi-dimensional rarefaction waves
Format : Talk at Waseda University
Author(s) :
Tianwen Luo (Tsinghua University)
Pin Yu (Tsinghua University)
Abstract : We study the two-dimensional acoustical rarefaction waves under the irrotational assumptions. We provide a new energy estimates without loss of derivatives. We also give a detailed geometric description of the rarefaction wave fronts. As an application, we show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves. This is a joint work with Prof. Pin Yu in Tsinghua Univerisity.
[04947] Nonlocal traffic flow models
Format : Talk at Waseda University
Author(s) :
Thomas Hamori (University of South Carolina)
Yongki Lee (Georgia Southern University)
Yi Sun (University of South Carolina)
Changhui Tan (University of South Carolina)
Abstract : In this talk, I will discuss a family of traffic flow models. The conventional Lighthill-Whitham-Richards model is notorious for exhibiting finite-time shock formation for all generic initial data. To address this issue, I will introduce a family of nonlocal traffic flow models, which incorporate look-ahead interactions. These models can be derived from discrete cellular automata models. Interestingly, we will explore how the nonlocal slowdown interactions prevent the shock formation, under certain suitable settings. Furthermore, I will also discuss the extension of these nonlocal models to second-order traffic flow models.
