Graduate Seminar on PDE in the Sciences

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

The seminar takes place on Fridays, at 14:15 in SR 2.040.

Schedule

Past talks

• Friday February 02

  Speaker: Illia Karabash

  Title: Homogenization and nonselfadjoint spectral optimization for eigenvalues of Maxwell systems

  
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The homogenization of eigenvalues of non-Hermitian Maxwell operators will be considered with the help of the H-convergence method. The Maxwell system is assumed to be equipped with suitable m-dissipative boundary conditions, namely, with Leontovich or generalized impedance boundary conditions.
We found a wide class of impedance operators with the property that nonzero spectra of associated Maxwell operators are discrete. To this end, a new embedding theorem is obtained.
We prove the convergence of eigenvalues to an eigenvalue of a homogenized Maxwell operator under the assumption of the H-convergence of the material parameters, and as a by-product, obtain the existence of an eigenvalue-free region around zero.
This result is applied then to nonselfadjoint eigenvalue optimization problems stemming from Photonics. If time allow us, connections with (non-)unique continuation for Maxwell systems will be also discussed. The talk is based on a joint research with Matthias Eller.

• Friday January 26

  Speaker: Iulia Cristian

  Title: On an inhomogeneous coagulation model with a differential sedimentation kernel

  
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We study an inhomogeneous kinetic model that describes rain initiation times or the behavior of air bubbles in water. It involves a coagulation term with a so-called differential sedimentation kernel, which has homogeneity larger than one and vanishes on the diagonal. In the spatial variable it contains a transport term modeling the sedimentation of clusters. For the homogeneous version of the model we prove that there is instantaneous gelation, more precisely, that there is no mass conserving solution for any positive time.

For the inhomogeneous model on the other hand we prove the existence of mass conserving solutions for short times. Our proof also includes the case of sum-type kernels of homogeneity larger than one, for which non-existence of solutions (independently whether mass-conservation happens) has been proven in the homogeneous case.

This is based on a joint work with B. Niethammer and J. J. L. Velázquez.

• Friday January 12

  Speaker: Daniel Sánchez Simón del Pino

  Title: The general Grad-Rubin boundary problem for the MHS Equations

  
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In this talk I discuss the well-posedness for the classical MHS equations in some classes of bounded domains. I focus mostly in a set of boundary conditions that were suggested by Grad and Rubin in the 1950's, namely to prescribe the normal component of the magnetic field in the whole boundary of the domain and in addition, the tangential component in part of the boundary. After discussing the current known results for the particular case of the torus, we will explain how to extend these techniques to general bounded domains and even some unbounded situations.

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

• Friday December 15

  Speaker: Antonio Tribuzio

  Title: Energy barriers for boundary nucleation in a two-well model

  
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Shape-memory alloys are specific materials that, e.g. during a cooling process, change their crystalline structure. For this, their internal elastic energy is a multi-well functional acting on the deformation gradient. This trasformation is often initiated by the formation of a small inclusion of deformed material (nucleation).
In this talk, we study scaling laws for a double-well singularly-perturbed elastic energy in which the inclusion of deformed material is constraint in the halfspace and has prescribed volume.
This problem is a variant of the isoperimetric problem with an additional (nonlocal and anisotropic) bulk term given by the elastic energy. We will see how the relation between the anisotropy of the material and the constraint affects the scaling.
This is a joint work (in preparation) with Konstantinos Zemas.

• Friday December 8

  Speaker: Bernhard Kepka

  Title: Overview of the Nash-Moser method for small divisor problems

  
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Various PDEs can be solved using perturbative methods, assuming a good enough approximate solution is already known. The construction of a solution then often relies on finding an inductively defined sequence of approximate solutions. Such a sequence might be given for instance via the linearised problem.

We give an overview of the Nash-Moser method to tackle problems in which solving the linearised problem involves a loss of regularity. Such problems in principle do not allow an application of a standard fixed point theorem. This occurs in particular for small divisor problems. As a toy example we discuss a conjugacy problem.

If time permits we give a short account of the application of the Nash-Moser approach to a work in progress with Juan J. L. Velázquez on steady states of the Euler-Poisson equations in three dimensions.

• Friday, November 24 at 14:15 in Lipschitz Saal
• joint seminar with RG Analysis and PDE

Speaker: Geoffrey Lacour

Title: Energy solution for a non-Newtonian Stokes-Transport problem


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In this talk, we will present the study of a model for particle suspensions in a non-Newtonian Ostwald-DeWaele fluid with a potentially degenerate viscosity coefficient.

The analysis of problems associated with such systems is a very active research topic, the source of many recent results, particularly in the case of particles that sediment in a Newtonian fluid (see, for example, the works of D. Cobb, R. Höfer, A. Mecherbet, R. Schubert, F. Sueur). We will consider the case where particles are suspended in a non-Newtonian fluid. From a mathematical point of view, this is characterized by a Stokes-Transport equation with the particularity that the Stokes equation is nonlinear, the viscosity term being expressed as a p-Laplacian for the symmetrized gradient. After briefly contextualizing the problem, we will present our result: the existence of global weak energy solutions.

In order to show that we do define an active scalar equation, i.e. one in which the relative density of the particles suspended in the fluid gives meaning to a solution of the system through an inverse mapping, it is then necessary to use monotonicity methods in conjunction with techniques derived from DiPerna-Lions theory to establish the existence of suitable weak solutions. We will therefore present the main ideas for establishing the existence of such ones, and end the talk by presenting some associated open problems.

This is a work done in collaboration with D. Cobb (University of Bonn).

Speaker: Ruoyuan Liu

Title: Local well-posedness of a quadratic nonlinear Schrödinger equation on the two-dimensional torus


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In this talk, I will present results on local well-posedness of the nonlinear Schroedinger equation (NLS) with the quadratic nonlinearity |u|^2, posed on the two-dimensional torus, from both deterministic and probabilistic points of view. For the deterministic well-posedness, Bourgain (1993) proved local well-posedness of the quadratic NLS in H^s for any s > 0. In this talk, I will go over local well-posedness in L^2, thus resolving an open problem of 30 years since Bourgain (1993). In terms of ill-posedness in negative Sobolev spaces, this result is sharp. As a corollary, a multilinear version of the conjectural L^3 -Strichartz estimate on the two-dimensional torus is obtained. For the probabilistic well-posedness, I will talk about almost sure local well-posedness of the quadratic NLS with random initial data distributed according to a fractional derivative of the Gaussian free field. I will also mention a probabilistic ill-posedness result when the random initial data becomes very rough. The first part of the talk is based on a joint work with Tadahiro Oh (The University of Edinburgh).

• Friday November 17

  Speaker: Richard Schubert

  Title: Generation of graph structure and long time behaviour

  
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We will discuss initial and long-time behaviour of the Mullins-Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. In particular we will consider initial data that are finite L^1 perturbations of the planar profile, and see how graph structure is generated, starting from a surface that neither needs to be a graph, nor even be connected (Ostwald ripening). We will gain some intuition by looking at the linearized situation and the scale invariant quantities. Based on joint work with Felix Otto and Maria G. Westdickenberg.

• Friday October 27

  Speaker: Juan Velázquez

  Title: Some mathematical problems arising in the theory of chemical networks

  
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In this talk I will present some mathematical questions motivated by problems of biochemistry and that I think that suggest new interesting research directions. If there is time, I will talk about three problems, namely:


1. What are the precise mathematical conditions under which a chemical system can be approximated by models in which the detailed balance condition fails?


2. How to determine if the detailed balance condition holds in a biochemical system (with many components) from measurements of the chemical concentrations of a few substances?


3. There are some particular chemical networks that discriminate between different classes of molecules with different affinities for some receptors with much more accuracy than the one predicted by equilibrium statistical physics. However, this is achieved in open systems, and spending energy. It is possible to prove mathematical theorems yielding the maximum discrimination ratio for two types of molecules in terms of the amount of energy spent in the process?

• Friday October 13

  Speaker: Eugenia Franco

  Title: Description of Chemical Systems by means of Response Functions

  
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In this talk, we introduce a formalism to describe the response of a biochemical system to a signal. Due to the large number of substances involved in the reactions, biochemical systems are, at a practical level, difficult to model in detail via a system of ODEs.
This motivates us to study, instead, only the interactions between different parts of biochemical systems via linear renewal equations. The response functions (or kernels) of the renewal equations summarize the dynamics inside each part of the system.
We will discuss some of the main properties of these renewal equations, such as the long time behaviour of their solutions.
As an example of model to which the formalism of response functions can be applied, we will see the Hopfield model of kinetic proofreading.
Finally, we will sketch how to extend the formalism of response functions to non-linear problems.

This work is based on a collaboration with B. Kepka and J.J.L Velázquez.