Graduate Seminar on PDE in the Sciences

Organizers: B. Niethammer, J.J.L. Velázquez, I. Cristian

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Schedule

June: Show Schedule

May: Show Schedule

April: Show Schedule

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Past talks

• Friday June 30

  Speaker: Théophile Dolmaire

  Title: Derivation of the Boltzmann equation: a first variation of Lanford’s theorem in the disk

  
  Show Abstract

The Boltzmann equation constitutes a tremendously efficient model to describe rarefied gases. On the other hand, the equation encodes a priori paradoxical behaviours, such as an irreversible trend to tend to equilibrium for the solutions. Such a behaviour is indeed an apparent paradox, since the equation aims to describe a fluid that is assumed to be constituted of elementary particles, evolving according a reversible model. Therefore, how can irreversibility arise from a reversible, microscopic behaviour?

Lanford’s pioneering result demonstrated that this paradox is not the consequence of an erroneous computation, but can be rigorously proved: irreversibility at a macroscopic scale can arise from irreversible microscopic models (here, following Lanford, we will model the particles as hard spheres).

In this presentation, we will describe the proof of an adaptation of Lanford’s result, in the case when the domain is the unit disk (we consider a 2-dimensional problem). Such a problem remained open (no rigorous adaptation is known in the case of a general domain). In particular, the proof of Lanford relies on an explicit expression of the solutions of the BBGKY hierarchy (the sequence of equations satisfied by the marginals of the distribution function of the system of hard spheres). Such an expression enjoys a natural, geometrical interpretation, based on the so-called pseudo-trajectories. The study of these pseudo-trajectories and in the particular the control of the recollisions will be the central question we will tackle.

This is a work in progress, realized with Chiara Saffirio (Universität Basel).

• Friday June 23

  Speaker: Richard Schubert

  Title: A variational approach to generalized-Newtonian Navier-Stokes

  
  Show Abstract

I will discuss an approach to obtain Leray-Hopf solutions for generalized-Newtonian Navier-Stokes equations, i.e. where the viscous stress is allowed to depend nonlinearly on the strain while stress and strain still have to align. We construct solutions as limits of a sequence of minimizers of space-time (so-called WIDE) functionals. In particular we will consider viscosity power-laws with low implied integrability where weak solutions cease to be energy solutions ($p<\frac{11}{5}$ in $d=3$). Here, I will try to show how the Newtonian case is special and how it nevertheless embedds in the general theory by truncation arguments. This is work in progress together with Christina Lienstromberg, Michael Ortiz and Stefan Schiffer.

• Friday June 09

  Speaker: Roberto Maturana

  Title: Decay of the Green's function of the fractional Anderson model and connection to long-range SAW

  
  Show Abstract

We prove a connection between the Green's function of the fractional Anderson model and the two point function of a self-avoiding random walk with long range jumps, adapting a strategy proposed by Schenker in 2015. This connection allows us to exploit results from the theory of self-avoiding random walks to improve previous bounds known for the fractional Anderson model at strong disorder. In particular, we enlarge the range of the disorder parameter where spectral localization occurs.

Joint-work with Prof. Disertori (Uni-Bonn) and Prof. Rojas-Molina (CY Cergy Paris Université)

• Friday June 02

  Speaker: Peter Gladbach

  Title: Variational interacting particle systems

  
  Show Abstract

Bernhard and I treat mean-field interacting particles that together minimize an action functional. We show that critical points of the boundary-value problem, where every particle has a prescribed start and end point, are precisely weak solutions of certain Vlasov equations. We show that there is a large class of action functionals that do not admit minimizers and construct their relaxations.

• Friday May 19

  Speaker: Nicola Miele

  Title: Homoenergetic flows for the Linear Boltzmann Equation

  
  Show Abstract

Homoenergetic solutions were introduced by Truesdell in an attempt to give exact solutions for the Boltzmann Equation. They represent a middle ground between the standard BE and the spatially homogeneous case. Despite this simplification they still present hard challenges to understand the behaviour of these solutions. The long-time behaviour is strongly dependent on the homogeneity of the collision kernel. In this seminar an overview of the topic is given with particular emphasis on the so collard hyperbolic-dominated case in which some conjectures are presented.

• Friday May 05

  Speaker: Daniel Sánchez-Simón del Pino

  Title: Well posedness of MHS equation with Grad-Rubin boundary conditions

  
  Show Abstract

In this talk, we investigate the well-posedness of the Magneto Hydrostatic Equations (MHS) with Grad Rubin boundary conditions. These are a family of equations that arise when taking the zero velocity regime of the Magneto Hydrostatic Dynamics (MHD). This problem has applications in plasma confinement experiments as well as solar wind simulations. We show how to adapt the ideas that Alber developed for the steady state Euler's equation in order to obtain perturbative solutions of MHS around an initial state. We focus on the 2D setting developed by Velázquez and Alonso-Orán. If time allows, we will present the current state of our generalization for the 3D situation and possible simplifications of the original proof by means of pseudodifferential operators.

• Friday Apr 28

  Speaker: Illya Karabash

  Title: Discrete and continuous 4-D dynamical systems arising in resonance optimization and extremal synthesis

  
  Show Abstract

It is planned to show how the continuous dynamical system of the Euler-Lagrange eigenproblem for optimal resonances generates a discrete 4-D dynamical system corresponding to switch points, and to discuss how these systems are related to patterns of optimal structures. If time allows us, we discuss also the general transition from optimal control and the Boltyansky-Pontryagin version of Maximum Principle to extremal synthesis and related continuous and discrete dynamical systems. The talk is partially based on the joint paper with Herbert Koch and Ievgen Verbytskyi.

• Friday Apr 14

  Speaker: Elena Demattè

  Title: The diffusion equilibrium approximation of the radiative transfer equation

  
  Show Abstract

We study the situation where a body interacts with radiation only. We consider the boundary value problem for the stationary radiative transfer equation under the assumption of the local thermodynamic equilibrium. We study the diffusion equilibrium approximation in the absence of scattering. We consider constant absorption coefficient independent of the frequency $\nu$ (the so-called Grey approximation) and the limit when the photons' mean free path tends to zero, i.e. the absorption coefficient tends to infinity. We show that the function $u$ of the temperature defined by the Stefan-Boltzmann law $u=T^{4}$ solves in the limit a Laplace equation where the boundary value can be determined uniquely in terms of the original boundary condition. We derive formally with the method of matched asymptotic expansions the boundary condition for the limit problem and we prove it rigorously with a careful analysis of some non-local integral operators. This is a joint work with Juan J.L. Velázquez.

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