Titles and Abstracts of Talks
Craig Allen
Title: Mathematical modeling of dialectical emergent hybrid regimes in social-ecological systems
Abstract: In traditional resilience research, ecosystems are often framed in terms of distinct “alternative regimes,” such as clear-water and turbid-water regimes in shallow lakes. These regimes are viewed as stable configurations that are either maintained by feedback loops or disrupted by external forces, leading to shifts from one regime to another. While this dichotomy is useful, it oversimplifies the dynamic interplay between regimes and neglects the potential for emergent hybrid regimes - novel ecological configurations that synthesize features of both opposing regimes while transcending their limitations. We seek to integrate the dialectical nature of ecosystem dynamics into a mathematical framework that goes beyond the conventional binary paradigm. It discusses how mathematical frameworks may provide powerful tools to analyze such dialectical systems and their emergent properties. Potential key approaches include: 1. Coupled differential equations, which capture interactions between opposing states and their contributions to hybrid dynamics; 2. Potential landscapes to visualize system stability and transitions, identifying tipping points and the conditions under which hybrid regimes emerge as stable configurations; 3. Stochastic modeling that incorporates randomness and external drivers, assessing the robustness of hybrid states under environmental variability; and 4. Process-relational models to dynamically track the evolution of emergent properties over time, integrating system memory and feedback into hybrid regime formation. These methodologies collectively offer a robust framework for exploring the dynamics, resilience, and management of ecosystems, moving beyond traditional dichotomies to a nuanced understanding of emergent hybrid regimes and their role in navigating ecological complexity.
Hakima Bessaih
Title: Numerical schemes for stochastic 2D Benard-Boussinesq equations
Abstract: We implement a semi-implicit time Euler scheme for the two-dimensional Benard-Boussinesq model on the torus. We prove a rate of convergence in probability of order (1/2)- for a multiplicative noise; this relies on moment estimates in various norms for the processes and the scheme. In case of an additive noise, due to the coupling of the equations, provided that the difference on temperature between the top and bottom parts of the torus is not too big compared to the viscosity and thermal diffusivity, a strong polynomial rate of convergence (almost 1/2) is proven for both the velocity and the temperature. These rates in both cases are like that obtained for the 2d stochastic Navier-Stokes equation.
Zachary Bradshaw
Title: An introduction to the AOT data assimilation framework
Abstract: In this talk we will discuss data assimilation, its role in forecasting as well as an approach developed circa 2014 by Azouni, Olson and Titi which is amenable to mathematical analysis.
Aynur Bulut
Title: Issues of instability and blowup in supercritical dispersive PDE
Abstract: TBA
Elizabeth Carlson
Title: Understanding large-time behavior using data assimilation
Abstract: One of the fundamental challenges of accurate simulation of turbulent flows is that initial data is often incomplete, which for said flows is a strong impediment to accurate modeling due to sensitive dependence on initial conditions. A continuous data assimilation method proposed by Azouani, Olson, and Titi in 2014 introduced a linear feedback control term to dissipative systems, giving a simple rigorous deterministic method by which to understand the underpinnings of more complex data assimilation algorithms. In this talk, we will focus on how the AOT algorithm and modifications can be improved by knowledge of the dynamics, as well as how the AOT algorithm can yield new insights into the large time behavior of said dynamical systems.
Mimi Dai
Title: Ill-posedness and singularity formation for electron MHD
Abstract: We will discuss some constructions for the electron MHD which either produce ill-posed solutions or solutions that develops finite time singularity.
Tarek M. Elgindi
Title: TBA
Abstract: TBA
Aseel Farhat
Title: TBA
Abstract: TBA
Gung-Min Gie
Title: Viscous Layers in Fluid Mechanics: Singular Perturbations, Analysis, and Computation
Abstract: Singular perturbation problems in PDEs, where a small parameter multiplies the highest-order derivatives, lead to challenges in analysis and computation due to the emergence of distinct ”fast” and ”slow” scales. These solutions often exhibit sharp transitions, such as boundary or interior layers, while remaining smooth elsewhere. In fluid mechanics, the Navier-Stokes equations for viscous flows can be treated as a singular perturbation of the inviscid Euler equations, with viscosity as the small parameter. This talk explores recent progress in understanding viscous layers in the Navier-Stokes equations and related systems. We analyze the struc- ture of boundary layers, the vanishing viscosity limit, and the role of correctors in refining theoretical results. Additionally, we discuss numerical methods and machine learning tailored for slightly viscous fluid equations, focusing on how viscous layer correctors enhance accuracy and efficiency. This work demonstrates the interplay between rigorous analysis and computational approaches in fluid mechanics.
Weizhang Huang
Title: A regularization approach to parameter-free preconditioning for the efficient iterative solution of singular and nearly singular Stokes, elasticity, and poroelasticity flows
Abstract: While Schur complement preconditioning has been widely studied for sad- dle point systems, challenges remain when coming to singular and nearly singular systems that arise from Stokes flows and nearly incompressible elasticity and poroe- lasticity flows. For such systems, existing studies focus on the development of pre- conditioners spectrally equivalent to the underlying system. Those preconditioners are effective by design; however, they are not efficient in general since they are slated to solve nearly singular systems. In this talk we will present a new approach for developing effective and parameter-free block Schur complement preconditioners for those saddle point systems. A key of this approach is to regularize original systems with inherent identities and construct preconditioners based upon the regularized systems. It will be shown these preconditioners are straightforward to construct and implement. Moreover, bounds on the eigenvalues of the preconditioned systems will be derived. The convergence of MINRES and GMRES applied to those systems will be analyzed and shown to be independent of locking parameters and mesh size. Numerical results in 2D and 3D will be presented.
Hao Jia
Title: Relaxation mechanisms in incompressible fluid flows and related models
Abstract: In this talk we will present several mechanisms for asymptotic stability in the two dimensional incompressible fluid equations, including inviscid damping, vorticity depletion, and enhanced dissipation. We will also discuss an important stability mechanism due to the balance between convection and vorticity stretching in the De Gregorio model for the three dimensional incompressible Euler equations. These stability mechanisms play an essential role in the long time behavior of smooth solutions to incompressible fluid equations, and have been used to prove asymptotic stability of important steady solutions. Numerical results and open questions about dynamical behavior of general solutions will also be presented.
Huynh Manh Khang
Title: Sparsity of Fourier mass of passively advected scalars in the Batchelor regime
Abstract: We propose a general dynamical mechanism that can lead to the failure of the Batchelor’s mode-wise power spectrum law in passive scalar turbulence and hyperbolic dynamics, while the cumulative law remains true. Of technical interest, we also employ a novel method of power spectral variance to establish an exponential radial shell law for the Batchelor power spectrum. An accessible explanation of the power spectrum laws via harmonic analysis is also given.
Evan Miller
Title: Finite-time blowup for the Fourier-restricted Euler equation
Abstract: In this talk, I will discuss finite-time blowup for a model equation the Euler and hypodissipative Navier-Stokes equations, where, in addition to the divergence free constraint, the constraint space now requires the velocity to be supported at a specific set of Fourier modes. While the Helmholtz projection is replaced with a projection onto a more restrictive constraint space, the nonlinearity is otherwise unaltered, and the divergence free constraint still holds. I will also briefly discuss the dynamics of the full Euler equation under this set of symmetries.
Weinan Wang
Title: Global well-posedness and stability for some fluid equations
Abstract: In this talk, I will talk about some recent well-posedness and stability results for two incompressible fluid equations. More precisely, I will first discuss a global stability result for the 3D Navier-Stokes equations. When the Navier-Stokes is coupled with the magnetic field in the magnetohydrodynamics (MHD) system, solutions near a background magnetic field are shown to be always global in time. The magnetic field stabilizes the fluid. If time permits, I will discuss some open problems. This is based on joint works with Bradshaw, Feng, and Wu.
Xukai Yan
Title: Homogeneous solutions of stationaryincompressible Navier-Stokes equations.
Abstract: Homogeneous solutions play an important role in the study of fluid equations. This talk focuses on homogeneous steady states of 3D incompressible Navier-Stokes equations (NSE) with an isolated singularity or finite singular rays. I will first talk about the classical Landau solutions, which are a family of explicit solutions with one singularity at the origin, discovered by Landau in 1944. Sverak proved that all (-1)-homogeneous solutions of the 3D stationary NSE that are smooth on the unit sphere must be Landau solutions. I will then talk about some recent study on (-1)- homogeneous steady states of NSE with singular rays in several aspects, including their existence and classification, as well as their singularity and stability behavior. I will also talk about homogeneous steady states of other fluid equations, such as Euler equations and Boussinesq equations.
Titles of poster sessions
- Pan Anping, "Variational Principe and Lagrangian-Eulerian Formulation of Hydrodynamic Equations"
- Anna William Ebo, "Studying Retinal Detachment Progression Using an Immersed Boundary Method"
- Richard Galen, "Infinitesimal Generator of a Multilayered Poroelastic System with Stokes Flow"
- Kwon Hyunwoo, "Nonstationary Stokes equations on a domain with curved boundary under slip boundary conditions"
- Burton James, "High-fidelity, front-tracking based simulations of the Raleigh Taylor instability with adaptive mesh refinement"
- Dalben C. B. Juliane, "On the long-time behavior of 2D stochastic hydrostatic Navier-Stokes equations"
- Luan Kunhui, "Critical thresholds in pressureless Euler-Poisson-Alignment System"
- Mu Yuhao, "Qualitative properties of a nonlinear fluid-structure interaction model"
- Prabin Sherpaili, "Pressure drop in insect wing"
- Pallock Rumayel, "The dynamical systems approach to active nematics using exact coherent structures"
- Aitzhan Sultan, "Bifurcation theory for coordinate free model of flame fronts"
- Walker Elliott, "Characterization of Near-surface Wind Profiles Based on Atmospheric Thermal Stability and Terrain Complexity"
- White Nick, "Data Assimilation in Large Eddy Simulation: Addressing Model-Observation Mismatch from Navier-Stokes Data"
- Angel Naranjo, `"Nonlinear Dynamics and Symmetries in Active Fluids"