Graduate Seminar on PDE in the Sciences

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

Reserved talks

Previous weeks

• Friday Feb 10 ♥ Special seminar

• First speaker

  Jens Scholten

• Second speaker

Lorena Pohl

Speaker: Jens Scholten

Title: Two component Becker-Döring Equations


Show Abstract

The Becker-Döring equations constitute the simplest coagulation-fragmentation system, where interaction between clusters can only occur through monomers (i.e. size one particles). The system is very well understood and can be used to model condensation in over saturated vapors. In the talk we consider a variant of the equation, where two different components (e.g. water and acid molecules) are present. We will briefly discuss existence, uniqueness and conservation of mass in the system. Then we discuss steady states of the system and the resulting challenges for the prediction of the long time behaviour.

Speaker: Lorena Pohl

Title: A Becker-Döring model with a broken chain of coagulation


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We consider a variation of the Becker-Döring equations with the first coagulation coefficient equal to zero. We show convergence to equilibrium for power-law coagulation and fragmentation rates and obtain a loss of mass in the limit $t\rightarrow \infty$ depending on the initial mass and the relative strengths of the coagulation and fragmentation processes. In the case of linear rates, we further show that large clusters evolve in a self-similar manner at large times by comparing limits of appropriately rescaled solutions in different spaces.

• Friday Jan 20

  Speaker: Iulia Cristian

  Title: Coagulation equations for non-spherical clusters

  
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In this talk, we investigate the long-time asymptotics of a coagulation model which describes the evolution of a system of particles characterized by their volume and surface area. The aggregation mechanism takes place in two stages: collision and fusion of particles (see Figure). We assume the coagulation kernel has a weak dependence on the area variable. We discover that the long-time analysis of the system is strictly related to the chosen fusion rate. We prove existence of self-similar profiles for some choices of the functions describing the fusion rate for which the particles have a shape that is close to spherical. On the other hand, for other fusion mechanisms, we show that the particle distribution describes a system of ramified-like particles.

If time allows, we will talk about how we are able to recover the standard coagulation equation in the case of fast fusion.

This is joint work with J. J. L. Velázquez.




Collision mechanism (left). Fusion mechanism (right)

• Friday Dec 16

  Speaker: Sebastian Hensel

  Title: Interface area minimization of planar networks of branched interfaces

  
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The concept of paired calibrations due to Lawlor and Morgan (Pacific J. Math., 166, 1994) provides a particularly elegant tool for proving global area minimization of a network of branched interfaces as appearing in multiphase materials. In this talk, I will present a result on local interface area minimization for a class of stationary points of the interface energy which, due to Kinderlehrer and Liu (Math. Models Methods Appl. Sci., 11, 2001), occur as long-time asymptotic limits of multiphase mean curvature flow. This class contains continuous one-parameter families of stationary points which are not global minimizers. In particular, paired calibrations do not exist for those and consequently, minimality properties for such networks remained an open problem. Our result is based on a new concept of paired local calibrations allowing to overcome the difficulties associated with the above mentioned degeneracy of the energy landscape. This is joint work with Julian Fischer, Tim Laux and Theresa Simon.

• Friday Dec 09

  Speaker: Eugenia Franco

  Title: Long-term behavior of the solutions of the coagulation equations with injection that do not have stationary solutions

  
  Show Abstract

Coagulation equations including a source term are a generalization of Smoluchowski's coagulation equations that is extensively used to model the dynamics of the aerosols in the atmosphere.
The existence/non-existence of non-equilibrium stationary solutions for these equations has been studied in [1].
It has been proven there that the existence of a stationary solution is determined by the assumptions on the coagulation kernel. More precisely, kernels that strongly promote the interaction of particles of very different sizes lead to the non-existence of stationary solutions. It is, therefore, interesting to understand the long-term behavior of the solutions of the coagulation equation with a source in this case.

In this talk, we discuss the existence/non-existence of self-similar solutions for coagulation equations including a source term for non-gelling, homogeneous coagulation kernels that strongly promote the coagulation of particles of very different sizes and for which a stationary solution of the coagulation equation with source does not exist.
We will see that, in this case, the coagulation equation with source can be approximated, for large sizes and large times, by a coagulation equation with a non-linear transport term.
We provide assumptions on the kernels that determine the existence/non-existence of self-similar solutions for this class of coagulation equations with a transport term.
In the case of existence, we will see that the self-similar solutions are zero near the origin, in contrast with what is seen for self-similar solutions of the classical Smoluchowski's coagulation equation.

Based on a collaboration with I.Cristian, M.A.Ferreira and J.J.L Velázquez.

• Friday Dec 02

  Speaker: Bernhard Kepka

  Title: Rotating solutions to the incompressible Euler-Poisson equation

  
  Show Abstract

We consider an incompressible fluid body, together with the self-induced gravitational force, which is perturbed by a small external particle rotating around the configuration. The fluid body is assumed to be flat, i.e. two dimensional, which can be interpreted as a simplified model for a flat galaxy. Our goal is to construct solutions, which are associated with steady states in a rotating coordinate system, in a perturbative setting. To this end, we apply the Grad-Shafranov method as well as conformal mappings.

If time permits, we will also discuss the difficulties in the corresponding three dimensional problem.

This is joint work with Juan Velázquez and Diego Alonso-Orán.

• Friday Nov 11

  Speaker: Roberto Maturana

  Title: The decay of the Green’s function of the fractional Anderson model and connection to long-range SAW

  
  Show Abstract

The Anderson model serves to study the absence of wave propagation in a medium in the presence of impurities. In this talk, we shall introduce the fractional Anderson model, which has the particularity that the kinetic energy of an electron is given by the discrete fractional Laplacian. This poses some challenges since the operator is non-local. To prove the absence of propagation of the electrons, we will use the Method of the Fractional Moments. Moreover, we will establish a relationship between the decay of the Green’s function of the fractional Anderson model and long-range SAW.

This result is a generalization of Theorem 1 in "How large is large? Estimating the critical disorder for the Anderson model. Letters in Mathematical Physics, 105 (1), 1-9."

This is a joint-work with Prof. Disertori (Uni-Bonn) and Prof. Rojas-Molina (CY Cergy Paris Université).

• Friday Nov 04

  Speaker: Richard Schubert

  Title: The inertialess limit for sedimenting particles

  
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In this talk we investigate systems of particles immersed in a fluid and subject to gravitation. The particles interact with each other through the fluid in a very implicit way. I will show how, relying on previous results for mean-field limits of inertialess particles, a mesoscopic description of particles with inertia can be derived. In particular I will point out the connection to the Transport-Stokes (and maybe the Vlasov-Brinkmann) equations as mean-field limits. This work in progress (but close to the finish line) relies on the approximation of the system by a system with much more explicit interaction and a detailed understanding of the involved forces in comparison to the case of a single particle. This is based on joint work with Richard Höfer (Paris).

• Friday Oct 28

  Speaker: Théophile Dolmaire

  Title: The Boltzmann equation: its main properties, and an introduction to its rigorous derivation

  
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The description of diluted gases relies on the Boltzmann equation. Despite its fascinating efficiency to encode complex physical behaviors we can observe at our scale, an important paradox arises from this model. Indeed, this equation can be formally obtained from a reversible description of the matter at a microscopic level, assuming that the atoms behave like billiard balls, and evolve according Newton’s laws. On the other hand, the behavior of the fluids verifying the Boltzmann equation is irreversible, since the solutions converge ineluctably towards some equilibrium states.

For this reason, one may wonder if the derivation that was performed is rigorous, and if the equation itself constitutes really a meaningful modelisation of reality. It is not until 1973 that this apparent paradox was properly addressed by Lanford, who provided the first rigorous derivation of the Boltzmann equation.

The Boltzmann equation has been introduced and its formal properties described along the two first talks, so for this third presentation, we will focus on the problem of the rigorous derivation. We will study the main tools involved in Lanford’s proof. Starting with a sequence of equations encoding the dynamics of a finite number of particles (the BBGKY hierarchy), we will see how it is linked with the Boltzmann equation, and we will present a function"al setting in which those objects make sense. Then, in this setting which allows to consider strong solutions of the hierarchies, we will prove the convergence of the solutions of the BBGKY hierarchy towards the Boltzmann hierarchy.

If time allows it, we will then investigate how the proof of Lanford can be extended to obtain a rigorous derivation of the Boltzmann equation in a domain with a boundary, with specular reflection as a boundary condition, which remained an open problem. We will start with the simplest case of a non-trivial domain: the half-space. Then, we will see how the proof dealing with the latter case can be modified for the setting of particles evolving around a general, two-dimensional convex obstacle. Finally, we will discuss a work in progress concerning the derivation for particles contained in the disk.

• Friday Oct 14

  Speaker: Jona Lelmi

  Title: A model for tumor growth with nutrients

  
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I will present a system of PDEs used as a simple model for tumor growth. The tumor cells are assumed to grow by eating nutrients which are supplied from the external environment. The nutrients are consumed at an exponential rate by tumor cells, and they are also assumed to diffuse according to a non-negative parameter D. We will describe a notion of weak solutions for the system of PDEs and we will see how one can make use of Wasserstein projections to construct them. When the model is assumed to have no nutrients diffusion (D=0), Kim et al. proved that the boundary of the tumor patch becomes regular after finite time, while numerical experiments show that for D>0 the tumor presents dendritic growth. It is then natural to ask whether some information is lost between the model with a small choice of D and with D=0. We will discover that within finite time the answer is no, at least in terms of Hausdorff distance. Based on joint work with Inwon Kim (UCLA).