Abstracts

Bei Hu, Notre Dame (Sep 27, 2022)

Title: A Free Boundary Problem for modeling Plaques in the Artery – Recent progress

Abstract: Atherosclerosis is a leading cause of death worldwide; it originates from a plaque which builds up in the artery. We considered a simplified model of plaque growth involving LDL and HDL cholesterols, macrophages and foam cells, which satisfy a coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. In an earlier work (with Avner Friedman and Wenrui Hao) of an extremely simplified model, we proved that there exist small radially symmetric stationary plaques and established a sharp condition that ensures their stability. In our recent work (with Evelyn Zhao), we look for the existence of non-radially symmetric stationary solutions. The absent of an explicit radially symmetric stationary solution presents a big challenge to verify the Crandall-Rabinowitz theorem; through asymptotic expansion, we extend the analysis to establish a finite branch of symmetry-breaking stationary solutions which bifurcate from the radially symmetric solutions. Since plaque is unlikely to be strictly radially symmetric, our result would be useful to explain the asymmetric shapes of plaque. Our recent work (with Yaodan Huang, Xiaohong Zhang, Zhengce Zhang) extends to other possible shapes as well as more realistic modeling efforts.

Charis Tsikkou, West Virginia University (Oct 11, 2022)

Title: Amplitude Blowup in Radial Euler Flows

Abstract: We show that the full compressible Euler system admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous wave moving toward the origin. At time of focusing, the primary flow variables suffer amplitude blowup at the origin. The flow is continued beyond collapse and gives rise to an expanding shock wave. We verify that the resulting flow provides a genuine weak solution to the full, multi-d compressible Euler system. While unbounded radial Euler flows have been known since the work of Guderley (1942), those are at the borderline of the regime covered by the Euler model: the upstream pressure field vanishes identically (either because of vanishing temperature or vanishing density there). In contrast, the solutions we build exhibit an everywhere strictly positive pressure field, demonstrating that the geometric effect of wave focusing is strong enough on its own to generate unbounded values of primary flow variables. This is joint work with Helge Kristian Jenssen (PSU).

Sze-bi Hsu, National Tsing-Hua University, Taiwan (Nov 1, 2022)

Title: A Spatiotemporal Model of Drug Resistance in Bacteria with Mutations

Abstract: Motivated by the experiments of Roy Kishony et. ( Science 353, 2016) on the evolution of bacteria in a large , spatial structured environment . A spatiotemporal dynamics model is constructed to study the effects of mutations on the persistence and extinction of bacteria under antibiotic inhibition. We construct a mixed type Lyapunov functional to prove the global stability of extinction state and coexistence state for the case of forward mutation and forward-backward mutation respectively.

Robert De Jaco , NOST (Nov 8, 2022)

Title: Formation of Traveling Waves in Single-Solute Chromatography

Abstract: Recent developments in materials science have led to the discovery of new materials for efficient carbon capture by adsorption or chromatography. However, industrial implementation is slow due in part to limitations in the understanding of the relationship between adsorption equilibria (a material property) and solute movement (the mechanism of separation). Current understanding is restricted to regimes where adsorption is infinitely fast, or the column is infinitely long.
In this talk, I highlight the importance of quantifying the relationship between solute movement and adsorption equilibria. Using a Galilean transformation, an asymptotic analysis in fast adsorption is employed to identify the time scale for which traveling waves are dominant. Related transformations moving at a time-dependent speed, allow me to investigate the formation of traveling waves. When the concentration inlet to the column initially at steady state is changed by a step function, a time-dependent speed cannot explain the formation of the traveling wave. When the concentration is changed in a fast, but smooth, manner, however, the time-dependence of the velocity allows for the initial condition to be satisfied. A numerical approach sheds light on the formation of traveling waves from a step change in concentration. Finally, I highlight a few additional aspects of solute movement that are not fully understood. This presentation describes joint work with Anthony Kearsley.

Ivan Sudakow, Dayton (Nov 15, 2022)

Title: Complex bifurcations in Bénard–Marangoni convection

Abstract: We study the dynamics of a system defined by the Navier–Stokes equations for a non-compressible fluid with Marangoni boundary conditions in the twodimensional case. We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as compared to bifurcations in the classical Rayleigh–Bénard and Bénard–Marangoni systems with simple linear vertical temperature profiles. In terms of the Bénard–Marangoni convection, the obtained mathematical results lead to our understanding of complex spatial patterns at a free liquid surface, which can be induced by a complicated profile of temperature or a chemical concentration at that surface. In addition, we discuss some possible applications of the results to climate science.

Ryan Thompson, U. North Georgia (Nov 22, 2022)

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Vinent Calvez, CNRS Lyon (Jan 17, 2023)

Title: The Fisher infinitesimal model under convex stabilizing selection

Abstract: In a joint work with David Poyato (Univ. Granada) and Filippo Santambrogio (Univ. Lyon), we exhibited a remarkable convexity structure associated with the Fisher infinitesimal model of genetic inheritance. As a by-product, we could prove uniqueness of the ground state together with long-time exponential convergence towards this state.

Shouhong Wang, Indiana (Mar 7, 2023)

Title: Einstein Equations, Dark Matter and Dark Energy

Abstract: The dark matter and dark energy phenomena are two unsolved mysteries in modern physics. In this talk, we intend to demonstrate that the presence of dark matter and dark energy requires the variation of the Einstein-Hilbert action be taken under energy-momentum conservation constraint. This gives rise to a new set of gravitational field equations, altering the Einstein equations with a new term analytically derived from the constraints. The new law of gravity we established shows that it behaves like the Einstein gravity in the solar system, and it has more attraction in the galactic scale (dark matter), and it becomes repulsive over very large scale (dark energy). I shall present various mathematical issues on the field equations. If time permits, I shall also discuss some mathematical issues on the structure of blackholes and the structure of the universe. This is joint work with Dr. Tian Ma.

Xueying Wang, Washington State (Mar 10, 2023)

Title: A reaction-advection-diffusion model of cholera epidemics with seasonality and human behavior change

Abstract: Cholera is a water- and food-borne infectious disease caused by V. cholerae. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of V. cholerae. We derive the basic reproduction number aquatic reservoir of V. cholerae. We derive the basic reproduction R0 for this system and then establish a threshold type result on its global dynamics in terms of R0. Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of R0 for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread. disease risk, and describe the asymptotic behavior of R0 for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread. (https://link.springer.com/article/10.1007/s00285-022-01733-3#citeas)

Dan Geba, Univ. Rochester (Mar 21, 2023)

Title: Unconditional well-posedness for Kawahara equations

Abstract: The Kawahara and the modified Kawahara equations are two nonlinear dispersive equations known to model shallow water waves. In this talk, we focus on their well-posedness theory, in particular the topic of unconditional uniqueness. We present arguments in this direction based on the method of normal form reductions. One of these results is based on joint work with Bai Lin.

Zhi-An Wang, Hong Kong Polytechnic University (Mar 28, 2022)

Title: Effects of the density-dependent dispersal on population dynamics

Abstract: Dispersal strategies, predation and competition are main determinants shaping the structure and functioning of ecological communities and maintaining the biodiversity. Most of existing models describing the predator-prey or completion dynamics employ the random dispersal strategy. However, biological species will more likely use non-random dispersal to optimize their ecological fitness in changing environments. Among other things, in this talk, we shall discuss how density-dependent dispersal (meaning that the dispersal of one species depends on the densities of other species), as a non-random dispersal strategy, may influence population dynamics in various ecological processes, and explore how it is selected to promote species coexistence and hence increase species richness and diversity.

Connor Mooney , UC Irvine (Apr 4, 2022)

Title: The anisotropic Bernstein problem

Abstract: The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8. The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications and offer important technical challenges. We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with Y. Yang.

Henrik Kalisch, U. Bergen, Norway (Apr 11, 2022)

Title: Existence and uniqueness for a system of conservation laws arising in magnetohydrodynamics

Abstract: The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics. It was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear in the whole phase space. As a consequence, certain Riemann problems have no weak solutions in the traditional Lax admissible sense. It was argued in Hayes and LeFloch (1996 Nonlinearity 9 1547-63) that in order to solve the system, singular solutions containing Dirac delta distributions along the shock waves might have to be used. Solutions of this type were exhibited in Kalisch and Mitrovic (2012 Proc. Edinburgh Math. Soc. 55 711-29) and Sarrico (2015 Russ. J.Math. Phys. 22 518-27), but uniqueness was not obtained. In this lecture, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and we show that admissible solutions are unique in this framework.

Kazuo Yamazaki, Texas Tech (Apr 18, 2022)

Title: Recent developments on probabilistic convex integration

Abstract: We review recent developments on convex integration applied on stochastic PDEs. Examples of equations include Euler equations (compressible or incompressible), Navier-Stokes equations, Boussinesq system, MHD system, power-law model, transport-diffusion equation, and surface quasi-geostrophic equations. Types of noise can be white in time (additive, linear multiplicative, non-linear), white in space (additive), or space-time white noise (additive). Typical results obtained are the existence of infinitely many (and hence non-unique, even in law) solutions at the level of probabilistically strong solution (i.e., adapted to the natural filtration generated by the given noise).

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Does the page seem familiar? I started using this template thanks to my late colleague Professor Ching-Shang Chou.