Session VII — Analysis and Applications of Partial Differential Equations

Tuesday, October 27 · 4:30–6:00 p.m.
Chair: Franziska Weber

Continuum limit for one-dimensional TMD-like heterostructures

Diyi Liu · Lawrence Berkeley National Laboratory

In bilayer moiré materials made from layered transition metal dichalcogenides (TMDCs), one of the layers is typically gapped at the Fermi level. Such materials are therefore generally modeled via effective moiré-scale continuum models only directly involving the ungapped layer for a specific energy level, with the gapped layer creating an effective moiré-scale potential. We rigorously justify this approach for systems of one-dimensional coupled chains with similar essential features by proving convergence of the dynamics of wave-packets spectrally concentrated at band edges to their continuum approximations.

Dissipative Evolutions in Metric Spaces

Lauren Conger · Stanford University

We propose an evolution variational inequality framework that generalizes the classical structure of evolutions in Hilbert spaces driven by monotone operators to the metric space setting. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. We provide conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme which generalizes the classical JKO scheme for single species gradient descent. We prove existence of discrete solutions to the VMS by combining dissipativity with a new notion of convexity along barycentric interpolations. Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity for general metric spaces.

Uniqueness and nonlinear stability of positive entire solutions in parabolic-parabolic chemotaxis models with logistic source on bounded heterogeneous environments

Issa Tahir Bachar · San Jose State University

This talk focuses on the asymptotic behavior of a parabolic-parabolic chemotaxis system with a logistic source in a bounded heterogeneous domain Ω. The equations are \(u_t = \Delta u - \chi\nabla\cdot(u\nabla v) + u(a_0(t,x) - a_1(t,x)u - a_2(t,x)\int_\Omega u)\) and \(\tau v_t = \Delta v - \lambda v + \mu u\), with homogeneous Neumann boundary conditions for \(u\) and \(v\). Under suitable parameter assumptions, the system has a unique positive entire solution \((u^*(t,x), v^*(t,x))\). Every nontrivial nonnegative initial population \(u_0 \in C^0(\bar{\Omega})\), with nonnegative \(v_0 \in W^{1,\infty}(\bar{\Omega})\), generates a global classical solution converging to \((u^*,v^*)\) in \(C^0\) uniformly over initial times \(t_0\).

Small Data Global Existence for a Class of Quasilinear Wave Equations in FLRW Spacetimes with Accelerating Expansion

John Cavanaugh · California State University, East Bay

Nonlinear wave equations are used to model waves in fluids and electromagnetic waves. Establishing small data global existence for nonlinear wave equations provides a theoretical basis for applied scientists’ models. Quasilinear wave equations, having forcing and damping terms depending on derivatives of the solution itself, tend to need more delicate analysis than linear wave equations. Previous results for semilinear wave equations show accelerating expansion of the background is a useful mechanism to have small data global existence. This work is a natural extension, showing expansion is still a viable mechanism in the quasilinear regime. These results can give insights into other nonlinear problems, especially geometric ones like the Einstein equations. Methods developed in FLRW spacetimes may generalize to similar problems in more interesting backgrounds, such as black hole backgrounds. We summarize our results, and we describe how the vector field method is used in their proof.

Perfect stable initialization for sinusoidal neural networks

Simon Kuang · UC Davis

The training of deep and wide neural networks depends sensitively on their initial weights: too large and the network blows up and becomes untrainable by first-order methods, too small and the network degenerates into a linear model. The conventional theory on stable initialization is based on asymptotic universality arguments and Gaussian approximations justified by the Central Limit Theorem to bound inter-neuron dependence. But all of this is unnecessary, we find, for neural networks with sine activation. We propose an initialization ensemble with the property that over the initialization distribution, every neuron is independent and identically distributed, resulting in a drastically simplified theory and exact equality guarantees for the Sobolev norm. Neural networks perform competitively with conventionally-derived initializations for neural representation learning.