Partial Differential Equations and their Applications Seminar
The seminar meets on Tuesdays at noon, in B3.02.
Term 1 - 25-26
No seminar
Title: Spectral analysis of the material-independent modes for the Helmholtz equation
Abstract: In this talk I will discuss the spectral analysis of a family of nonself-adjoint spectral problems defined by a homogeneous Helmholtz equation when using the permittivity/permeability as the eigenvalue. The study of such problems is motivated by the modal decomposition approach in computational electromagnetism, wherein the electromagnetic field scattered by a nanocavity subject to radiation losses (i.e. in an open system), can be described as an infinite sum using as basis functions the eigenfunctions of a spectral problem defined by the unforced Maxwell's equations. Owing to radiation conditions, this linear spectral problem is non-self-adjoint, which is at the source of numerical and theoretical difficulties. Furthermore, the problem is non-standard in that its eigenvalues both diverge and accumulate at multiple finite points. I will present a rigorous spectral analysis of these modes and show their completeness in a suitable Hilbert space.
This is a joint work with Anne-Sophie Bonnet-Ben Dhia (ENSTA Paris) and Cristophe Hazard (ENSTA Paris).
Title: On an inhomogeneous coagulation modle with a differential sedimentation kernel
Abstract:
Coagulation equations describe the evolution in time of a system of particles that are characterized by their volume. Multi-dimensional coagulation equations have been used in recent years in order to include information about the system of particles which cannot be otherwise incorporated. Depending on the model, we can describe the evolution of the shape, chemical composition or position in space of clusters.
In this talk, we focus on a model that is inhomogeneous in space and contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass-conserving solutions for a class of coagulation rates for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs.
This is based on a joint work with B. Niethammer and J. J. L. Velázquez.
TBA
Title: Global Convergence of Adjoint-Optimized Neural PDEs
Abstract:
Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks and solving the inverse problem of learning such terms from observed data in order to discover hidden physics. The resulting neural-network PDE model, being a function of the neural network parameters, can be calibrated to the available ground truth by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. These neural PDE models have emerged as an important research area in scientific machine learning. In this talk, we discuss the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity. Specifically, for a general class of nonlinear parabolic PDEs with a neural network embedded in the source term, we prove convergence of the trained neural-network PDE solution to the target data (i.e., a global minimizer). The global convergence proof poses a unique mathematical challenge that is not encountered in finite-dimensional neural network convergence analyses due to (i) the neural network training dynamics involving a non-local neural network kernel operator in the infinite-width hidden layer limit where the kernel lacks a spectral gap for its eigenvalues and (ii) the nonlinearity of the limit PDE system, which leads to a non-convex optimization problem, even in the infinite-width hidden layer limit (unlike in typical neural network training cases where the optimization problem becomes convex in the large neuron limit). The theoretical results are illustrated and empirically validated by numerical studies.
This talk is based on joint work with Justin Sirignano (University of Oxford) and Konstantinos Spiliopoulos (Boston University), see https://arxiv.org/abs/2506.13633 for the preprint.
Title: Stability of gravitational collapse
Abstract: In the Newtonian setting, a star is modelled as a spherically symmetric gas obeying the compressible Euler-Poisson system. In certain regimes, smooth initial data may give rise to blow-up solutions, corresponding to the collapse of a star under its own gravity, and such solutions have been rigorously constructed In recent years. In this talk, I will present the nonlinear stability of the simplest of these blow-up profiles, the Larson-Penston solution to the Euler-Poisson equations. This is based on joint works with Yan Guo, Mahir Hadzic, and Juhi Jang.
Title: Wave propagation in hydrodynamic stability
Abstract: The stability of shear flows in the fluid mechanics is an old problem dating back to the famous Reynolds experiments in 1883. The question is to quantify the size of the basin of attraction of equilibria of the Navier-Stokes equations depending on the viscosity parameters, giving rise to the so-called stability threshold. In the case of a three-dimensional homogeneous fluid, it is known that the Couette flow has a stability threshold proportional to the viscosity, and this is sharp in view of a linear instability mechanism known as the lift-up effect. In this talk, I will explain how to exploit certain physical mechanisms to improve this bound: these can be identified with stratification (i.e. non-homogeneity in the fluid density) or rotation (i.e. Coriolis force). Either mechanism gives rise to oscillations which suppress the lift-effect. This can be captured at the linear level in an explicit manner, and at the nonlinear level by combining sharp energy estimates with suitable dispersive estimates.
Title: Continuous data assimilation for compressible temperature driven fluids
Abstract: We consider the Navier-Stokes-Fourier system describing the evolution of a compressible temperature driven rotating fluid arising in meteorology. We show convergence of a continuous data assimilation (CDA) method in the regime of low Mach/high Rossby numbers. This is the first result on convergence of CDA method for a system that is not (known to be) well posed.
Title: The non-homogeneous Euler equations below the Lipschitz threshold
Abstract: The incompressible Euler equations are well-known to be globally well-posed in the case of space dimension $d=2$, both in the strong solutions framework and in the Yudovich framework. No results of that kind are known for the non-homogeneous (that is, density-dependent) incompressible Euler system. In this talk, we show that both problems (\textsl{i.e.}, global well-posedness and theory \textsl{\`a la Yudovich} for the density-dependent case) can be reduced to the study of a non-linear geometric quantity, which encodes the regularity of the velocity field along the level lines of the density. Such a geometric regularity places itself below the Lipschitz threshold.
Title: The cubic Dirac equation and its non-relativistic limit
Abstract: In this talk, I will present a scale-invariant global well-posedness and scattering theory for cubic Diracequations. It is based on a modular approach using critical adapted function spaces and bilinear Fourier restriction theory. As an application, we can study the non-relativistic limit, more precisely, convergence towards a system of cubic nonlinear Schrödinger equations. This is joint work with Timothy Candy.
Term 2 - 25-26
No Seminar
TBA
TBA
Title: From a Rayleigh Gas to Boltzmann and Fractional Diffusion Equations
Abstract: The fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where particles interact via short range potentials with support of size r and the background is distributed in space according to a Poisson process with intensity N and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as r tends to zero and N tends to infinity with N r^2 =c. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times [0,T], where T and inverse mean free path c can both be chosen as some negative rational power r^{-k}. Base on joint work with Theodora Syntaka.
Title: Weak solutions for compressible viscoelastic fluid models in three space dimensions
Abstract: We discuss global in time existence of weak solutions to compressible visco-elastic fluid models in three space dimensions. The first result concerns the situation with corrotational derivative in the extra stress tensor. Then, assuming additionally that the extra stress tensor has a particularly simple structure, the existence of weak solutions can be shown even in the situation when the stress diffusion is neglected which is often the case in applications.
The second result concerns Oldroyd-B type of model. It is known that in three space dimensions the Newtonian structure for the viscous part of the stress tensor is not enough to ensure the existence of weak solutions for arbitrarily large data. However, assuming the stress tensor of the power-law type it is possible to close the estimates and construct solutions provided the extra stress diffusion is present and the model of the viscous stress tensor provides bounded velocity divergence.
The result is a joint work with Yong Lu from University of Nanjing.
TBA
Title: Mathematical justification of a compressible bi-fluid system
Abstract: A recurring hypothesis for deriving models governing the evolution of mixtures of compressible fluids is that one can zoom-in to a scale known as the mesoscopic scale, where the constituents are completely separated by sharp interfaces and the continuum hypothesis is still valid for each constituent. Since at this mesoscopic scale, each constituent occupies its own domain, in order to obtain a closed model one must encode the interactions between the fluids at the level of the interfaces. Of course, the number of interfaces is, or becomes, too large for a model written at this scale to be practically useful. For this reason, one seeks to derive an effective model for the “mean flow”.
The aim of this talk is to propose a mathematical framework for obtaining a macroscopic description of a mixture of two viscous fluids. The analysis and results are carried out in one space dimension. The results that I will present were obtained in collaboration with Didier Bresch, Frédéric Lagoutière and Pierre Gonin-Joubert.
