Titles and Abstracts
John Ball.
Monodromy and nondegeneracy conditions in viscoelasticity.
For certain models of one-dimensional viscoelasticity, there are infinitely many equilibria representing phase mixtures. In order to prove convergence as time tends to infinity of solutions to a single equilibrium, it is necessary to impose a nondegeneracy condition on the constitutive equation for the stress, which has been shown in interesting recent work of Park and Pego to be necessary. The talk will explain this, and show how in some cases the nondegeneracy condition can be proved using the monodromy group of a holomorphic function. This is joint work with Inna Capdeboscq and Yasemin Şengül.
Katharine Gurski.
A vector-borne disease model with non-exponentially distributed infection and treatment stages.
Most epidemiological models for vector-borne disease assume exponentially distributed residence times in disease stages in order to simplify the model formulation and analysis. However, these models do not allow for accurate description of the interaction between drug concentration and pathogen load within hosts. To improve this, we formulate a model by considering arbitrarily distributed sojourn for various disease stages. The model formulation is presented using two proven equivalent approaches: integral equations and partial differential equations. Analysis of the model includes the existence of equilibrium solutions and stability, which are shown to be dependent on whether the control reproduction number is less or greater than 1. It is also shown that, when the general distributions are replaced by gamma distributions, the system of integral equations can be reduced to a system of nonlinear ODEs, which has some non-trivial characteristics which are only captured by non-exponential distributions for disease stages.
Richard Kollar.
Self-organization as an absorbing Markov chain.
We explore self-organization in a stochastic dynamical system using rigorous and numerical multiscale analysis. The system, at the microscale, describes lane-switching of individuals moving in opposite directions on a circular multi-lane track. At the mesoscale, some system variants are modeled as absorbing Markov chains, while others are not. However, at the macroscale, all variants reduce to an absorbing Markov chain model, ensuring self-organization within a finite time. Our findings show a constant upper limit for the mean time to self-organization regardless of the number of individuals in the system in variants with mesoscopic Markovian properties. That aligns with a simplified Poisson process model following the Mass action law. Conversely, non-Markovian variants on the mesoscale exhibit longer self-organization times, although numerical simulations indicate that some of them also achieve bounded self-organization times. We also identify a non-Markovian variant that demonstrates crowd formation tendencies, leading to prolonged time of self-organization. For another variant, we confirm linear growth in self-organization time observed in the literature for simple lane-switching models. Our study provides a framework for applying absorbing Markov chain theory in self-organization, particularly in systems involving conflict resolution mechanisms.
Chen-Chih Lai.
Free boundary problems of PDEs concerning thermal effects on the dynamics of a gas bubble in an incompressible fluid.
We study the thermal decay of bubble oscillation in an incompressible liquid. Attention will be directed towards the thermal decay of bubble oscillation, particularly examining the approximate model proposed by A. Prosperetti in [J. Fluid Mech. 1991]. This model exhibits a one-parameter manifold of spherical equilibria, parametrized by the bubble mass. We prove that the manifold of spherical equilibria is an attracting centre manifold against small spherically symmetric perturbations and that solutions approach this manifold at an exponential rate as time advances. Moreover, we show that, with viscosity and surface tension present, all equilibrium bubbles are spherically symmetric through an application of Alexandrov’s theorem on closed constant-mean-curvature surfaces. Furthermore, the manifold of spherically symmetric equilibria captures all regular spherically symmetric equilibrium.
We also examine the dynamics of the bubble-fluid system subject to a small-amplitude, time-periodic external sound field. We prove that this periodically forced system admits a unique time-periodic solution that is nonlinearly and exponentially asymptotically stable against small spherically symmetric perturbations. If time permits, I will discuss a work in progress on asymmetric dynamics of these models and future directions.
This talk is based on joint work with Michael I. Weinstein ([Arch. Ration. Mech. Anal. 2023], [Nonlinear Anal. 2024], and work in progress).
Dave Levermore.
Global Dynamics for the Kompaneets Equation.
The Kompaneets equation governs the evolution of a photon energy spectrum due to Compton scattering in a spatially homogeneous plasma. We prove some results concerning the long-time convergence of solutions to Bose-Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose-Einstein condensation. We provide two conditions that guarantee a photon loss occurs, and show that once a loss is initiated then it persists forever. We prove that every solution has a large-time limit that is a Bose-Einstein density that can be characterized in terms of the total photon loss. Additionally, we provide some results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is an $L^1$ contraction. None of these results impose a boundary condition at the zero-energy boundary.
Jian-Guo Liu.
Macroscopic Dynamics for Nonequilibrium Biochemical Reactions from a Hamiltonian Perspective.
Most biochemical reactions in living cells are not closed systems; they interact with their surroundings by exchanging energy and materials. At a mesoscopic scale, the quantity of each chemical can be modeled by random time-changed Poisson processes. Understanding macroscopic behaviors is facilitated by a nonlinear reaction rate equation that describes species concentrations. In the thermodynamic limit, the large deviation rate function from the chemical master equation is governed by a Hamilton–Jacobi equation. We decompose the general macroscopic reaction rate equation into an Onsager-type strong gradient flow, supplemented by conservative dynamics. We will also present findings on the large deviation principle and the importance sampling of transition paths that connect metastable states in chemical reactions.
Govind Menon.
Brownian motion on Riemannian manifolds.
The rigorous foundations of Brownian motion on Riemannian manifolds was developed in the 1970s. However, our understanding of this problem, in particular the interplay between the underlying metric and the Brownian motion has been considerably enriched by recent applications.
We will discuss low-regularity constructions of Brownian motions and a new formulation of the isometric embedding problem. This is joint work with Dominik Inauen (Leipzig).
Ryan Murray.
Singularity Formation for 2D Scale-Invariant Incompressible Euler Equation.
This talk will discuss recent work, with Tarek Elgindi and Ayman Said, on a special class of scale-invariant solutions to the 2D Euler equation. Scale-invariant solutions describe the motion of certain classes of singularities of the Euler equations, and can be seen as one attempt to model certain coherent vortex structures. These solutions obey a non-linear scalar equation on the circle, which admits local well-posedness for vorticities in $L^p$. Our results 1) give a characterization of the attractors of this system for bounded vorticities, and 2) demonstrate finite-time blowup for specific data in Lp. Subsequently, these "1D" solutions can be used to construct solutions of 2D Euler with $L^p$ vorticity for which analytic functions norms blow up in finite time. The ideas behind these results, and their motivation, connect directly with many of the topics I learned from Bob as a PhD student.
José Quintero.
On the stability of standing waves for a 1D-Benney-Roskes system.
In this talk, we establish the nonlinear orbital stability of standing waves (ground states) for the Benney-Roskes system with $N=1$, \begin{equation} \label{ZR/BR} \left\{ \begin{array}{rl} i \partial_t \psi +\epsilon \partial_z^2 \psi&=- \sigma_1 \Delta_{\bot}\psi + \left(\sigma|\psi|^2 + W(\rho+ D\partial_z \varphi) \right)\psi, \\ \partial_t \rho + \sigma_2 \partial_z \rho &= -\Delta_{\bot}\varphi- \partial_z^2\varphi - D \partial_z(|\psi|^2),\\ \partial_t \varphi + \sigma_2 \partial_z \varphi &= - \frac{1}{M^2} \rho -|\psi|^2. \end{array}\right. \end{equation} This system describes the interaction of high-frequency and low-frequency waves in plasmas and magnetohydrodynamics, where we are using the notation $\mathbf x=(x, y, z)$ for $N=3$, $\mathbf x=(x, z)$ for $N=2$, $\Delta_{\bot}= \partial_x^2+\partial_y^2$ for $N=3$, and $\Delta_{\bot}= \partial_x^2$ for $N=2$. The model was first derived for D. Benney and G. Roskes in the context of gravity waves (Benney-Roskes-1969) and also for A. Rubenchik and V. Zakharov in the context of the interaction of spectral narrow high frequency wave packet of small amplitude with low-frequency acoustic type oscillations (Rubenchik-Zakharov-1972). For $N = 1$, $\partial_z=\partial_x$ and $u=\varphi_x$, the model appears in the context of Alfén waves propagation in plasma (see F. Oliveira-2003).
In the case $N=1$, we prove that the standing waves of the Benney-Roskes system are orbitally stable, by using the Lyapunov method, as done by M. Weinstein (Weinstein-1986) to establish stability for nonlinear Schr\"{o}dinger equation (NLS) and the generalization of the Kortewegde Vries equation (GKdV).
Walter Strauss.
Transverse Instability of Stokes Waves.
It was discovered in the 1960s, numerically and heuristically, that even very small Stokes water waves are unstable when subject to long-wave longitudinal perturbations. This is the modulational (Benjamin-Feir) instability. Around 1980 it was discovered numerically that they are also unstable when subject to transverse (3D) co-periodic.perturbations. I will present the first proof of this phenomenon. It is joint work with Huy Nguyen and Ryan Creedon.
What we prove is the spectral instability. The fluid is allowed to have either infinite or finite depth. A conformal mapping is used to fix the free boundary. The linearized operator is pseudo-differential. It is analytically expanded in the wave amplitude (epsilon) and the perturbation parameter (delta). Very lengthy calculations are required to find the unstable eigenvalue, which occurs at third order in epsilon. I will emphasize the methods and spare the audience the long calculations.
Juan J. L. Velázquez.
Specificity, sensitivity and speed in a stochastic kinetic proof reading model..
In this talk I will describe a stochastic version of the Hopfield-Ninio kinetic proof reading model. The Hopfield-Ninio model is a chemical network that was introduced in the 1970's to explain the extremely low number of errors in some biological processes like DNA transcription, mRNA translation and recognition of antigenes by the inmune system among others. The kinetic proof reading systems can be thought as error correcting mechanisms that reduce the amount of errors to levels much lower than the one that would be expected for systems at equilibrium. I will show how an asymptotic analysis of the model described above explains the extreme specificity of the Hopfield-Ninio model, i.e. the capability to discriminate between different ligands. Additional quantities like the amount of energy used by the network or the time required to yield a response will be also computed. If time allows, I will describe some properties of a spatially non-homogeneous coagulation model in which the presence of sedimentation terms eliminate the phenomenon of gelation that takes place for the spatially homogeneous version of the model. (This is joint work with Eugenia Franco).
Eugene Wayne.
Continuum approximations for the $\beta$-FPUT model..
It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. However, numerical experiments by Pace, Reiss, and Campbell show that for the $\beta$-FPUT model, the appropriate approximation equation is the modified-KdV equation, and in particular, the kink solutions of the mKdV equation play an essential role in the long-time asymptotics. We derive explicit approximation results for solutions of the $\beta$-FPUT using the mKdV as a modulation equation. In contrast to previous work, our estimates allow for non-localized solutions so that we can approximate kink solutions. These results allow us to conclude meta-stability results of kink-like solutions of the FPUT. This is joint work with Trevor Norton (Va Tech).
Michael I Weinstein.
Pseudo-magnetism and Landau Levels.
I will discuss the properties of waves in a periodic or deformed-periodic medium. A non-uniform deformation of a medium having the symmetries of a honeycomb tiling of the plane induces effective-magnetic and effective electric fields. I will first present a continuum (homogenized PDE) theory (joint work with J. Guglielmon and M. Rechtsman - Phys. Rev. A 103 2021). One may choose a deformation of the honeycomb medium which gives rise to a constant perpendicular effective magnetic field. For this choice, "Landau-Level spectrum", having "flat spectral bands" and very high density of states, are induced. This provides a mechanism to produce strong light-matter interactions. I will conclude with a discussion of very recent experimental confirmation of this theory (Barsukova et al. Nature Photonics, April, 2024).