PDE Research Group

PDE Seminar Fall 2023

Fridays, 3:30 — 4:20pm, Stevenson Center 1431 (in-person) or via Zoom (online). When possible, in-person talks will be live streamed.

Date: Wednesday, September 6, 4:10pm, 2023, location SC 1431. (Note the different date.)

  • Speaker: Benjamin Seibold, Temple University.
  • Title: Principled Mathematical Models for the Spotted Lanternfly Invasion.
  • Abstract: The Eastern United States are facing an invasive species: the spotted lanternfly, introduced in 2014 to Pennsylvania. Due to its ability to severely compromise lumber, grape, and crop production, it has been labeled “the worst invasive species to establish in the US in a century.” We showcase a collaborative effort to develop principled mathematical models for the lanternfly life cycle and its dependence on climatic conditions, with the goal to generate accurate predictions of the pest’s spread and establishment potential across the country. The developed models employ kinetic equations to capture the coexistence of agents at any development state. We discuss efficient numerical methods that enable detailed parameter studies, and outline how the methods can be combined with genetic drift as well as geo-spatial spread modeling.

Date: Friday, September 8, 2023.

  • Speaker: Tuoc Van Phan, University of Tennessee, Knoxville.
  • Title: On regularity theory in Sobolev space for a class of linear parabolic equations with degenerate coefficients.
  • Abstract: We discuss about a class of linear parabolic equations in the upper half space in which the leading matrix coefficients behave as x_d^\alpha, where
    alpha is in (0,2) and x_d is the vertical direction spatial variable. The main motivation to study this class of equations comes from the study for a class of degenerate viscous Hamilton-Jacobi equations. Under some weighted VMO condition on the coefficients, existence and uniqueness of solutions in suitable weighted Sobolev spaces are proved for equations in both divergence and non-divergence form. Some ideas and techniques in the proof will be represented. Discussion on related results, and open questions are also addressed. The talk is based on the joint works with Hongjie Dong (Brown University) and Hung Vinh Tran (University of Wisconsin – Madison).

Date: Friday, September 22, 2023.

  • Speaker: Andrew Krause, Durham University.
  • Title: Dynamical Systems Approaches and Interactive Visualisations for Pattern Formation.
  • Abstract: Natural patterns, such as those created during embryological development, can arise from enormously complex processes occurring across vast scales of space and time. A key scientific challenge is to conceptually map out these processes in terms of distinct mechanisms, and their interplay. Dynamical systems theory provides several tools for developing hypotheses regarding such processes, and for understanding the limitations of potential mechanisms.
    We will discuss the uses and limitations of linear and nonlinear analyses of reaction-transport models in the context of understanding problems of multiscale periodic patterning. A focus will be on understanding robustness and the ability for ‘generic’ models to exhibit different patterning behaviours, without having to quantify molecular details of a particular system. We will aim to demonstrate how these kinds of models and ideas can help generalize insights from specific systems and numerical simulations, while also discussing fundamental limitations to this kind of modelling. VisualPDE.com will be introduced as a tool to rapidly prototype simple models, as well as to teach and communicate aspects of PDEs more generally. We will end with a range of open problems, both technical and conceptual.
  • Zoom link: https://vanderbilt.zoom.us/j/95564288682

Date: Friday, October 6, 2023.

Date: Friday, October 13, 2023.

  • Speaker: Matthias Sroczinski, Konstanz University (Germany).
  • Title: Global existence and decay of small solutions for quasi-linear second-order uniformly dissipative hyperbolic-hyperbolic systems.
  • Abstract: We consider quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states are shown in space dimensions larger or equal to 3. The dissipation is characterized by algebraic conditions, previously developed by Freistühler and the speaker, equivalent to the uniform decay of all Fourier modes at the reference state. As a main technical tool para-differential operators are used. The result applies to recent formulations for the relativistic dynamics of viscous, heat-conductive fluids such as notably that of Bemfica, Disconzi and Noronha (2018.).
  • Zoom link: https://vanderbilt.zoom.us/j/92410579521

Date: Friday, October 27, 2023.

  • Speaker: Chuntian Wang, The University of Alabama.
  • Title: On the impact of spatially heterogeneous human behavioral factors on 2D dynamics of infectious diseases.
  • Abstract: It is well observed that human natural and social behavior have non-negligible impacts on spread of contagious disease. For example, large scaling gathering and high level of mobility of population could lead to accelerated disease transmission, while public behavioral changes in response to pandemics may reduce infectious contacts. In order to understand spatial characteristics of epidemic outbreaks like clustering, we formulate a stochastic-statistical epidemic environment-human-interaction dynamic system, which will be called as SEEDS. In particular, a 2D agent-based biased-random-walk model with SEAIHR compartments set on a two-dimensional lattice is constructed. Two environment variables are taken into consideration to capture human natural and social behavioral factors, including population crowding effects, and public preventive measures in the presence of contagious transmissions. These two variables are assumed to guide and bias agent movement in a combined way. Numerical investigations imply that controlling mass mobility or promoting disease awareness can impede a global-scale spatial population aggregation to form, and consequently suppress disease outbreaks. Importance of coordinated public-health interventions and public compliance to these measures are explicitly demonstrated. A mechanistic interpretation of spatial geometric traits in progression of epidemic transmissions is provided through these findings, which may be useful for quantitative evaluations of.

Date: Friday, Dec 1st, 2023.

  • Speaker: Patrick Heslin, Maynooth University (Ireland).
  • Title: Geometry of the Generalized Surface Quasi Geostrophic Equations.
  • Abstract: Arnold’s celebrated 1966 paper illustrates that the Euler equations of ideal hydrodynamics arise naturally, from the perspective of particle trajectories, as a geodesic equation on the group of smooth volume-preserving diffeomorphisms of the fluid domain equipped with a right-invariant metric corresponding to the fluid’s kinetic energy.

    In this talk we will investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation, a well-known model for the three-dimensional Euler equations.

    We will see that these equations can also be realised as geodesic equations on groups of diffeomorphisms. We will then illustrate precisely when the corresponding exponential map is non-linear Fredholm of index 0. Finally, we will examine the distribution of conjugate points in these settings via a Morse theoretic argument.