Seminar of Differential Equations and Numerical Analysis 2022-2023

Thursday 13/10/2022, 11:00-12:00

Room: E.324

Speaker: Grigorios Fournodavlos (University of Crete)

Title: Mathematical problems of General Relativity

Abstract: We will go over some classical topics in general relativity and discuss problems that are of interest to mathematicians. This is intended to be an introductory talk and anyone is welcome to attend.

Thursday 27/10/2022, 12:00-13:00

Room: E.324

Speaker: Nikolaos Athanasiou (University of Crete)

Title: Results on the formation of singularities for the 1-dimensional Relativistic Euler equations

Abstract: An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data set is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data (this being appropriately defined in terms of spatial gradients of the Riemann invariants). Our talk will address the question of whether this dichotomy persists for large data problems, possibly containing a far-field vacuum, at least for the system of the Relativistic Euler equations in (1+1) dimensions. I shall discuss results on both the isentropic and the non-isentropic cases.

The talk will be based on the following joint works with Shengguo Zhu (Shanghai Jiao-Tong) and Tianrui Bayles-Rea (Oxford): arXiv:1903.03355, arXiv:2106.07467.

Thursday 3/11/2022, 12:05-12:55

Room: E.324

Speaker: Ioannis Athanasopoulos (University of Crete)

Title: The Stefan problem

Abstract: The Stefan problem is one of the simplest free boundary problems of parabolic type. It is at the center of interest for more than 100 years. Although it is consider to be simple, its resolution is still far away even in its simplest form. We will focus our attention on the local regularity of the solution, as well as that of the free boundary, mainly in recent results.

Tuesday 15/11/2022, 14:05-14:55

Room: E.324

Speaker: Panos Karagiorgos (University of Crete)

Title: Ημικλασική Ανάλυση της Εξίσωσης Schrödinger Φασικού Χώρου

Abstract: Παρουσιάζουμε δύο μεθόδους κατασκευής ασυμπτωτικών λύσεων του ημικλασικού προβλήματος Cauchy για την εξίσωση Schrödinger φασικού χώρου, για αρχικά δεδομένα Lagrange φασικού χώρου και μικρά χρονικά διαστήματα. Η πρώτη μέθοδος πρόκειται για κατασκευή ασυμπτωτικής λύσης μέσω της κατασκευής ημικλασικού διαδότη, στη βάση της ανισοτροπικής ημικλασικής χρονικής εξέλιξης κυματοδεσμών Gauss. Η δεύτερη μέθοδος πρόκειται για κατασκευή ασυμπτωτικής λύσης μέσω της Μιγαδικής Θεωρίας WKB, στη βάση κανονικού συστήματος και ροής χαρακτηριστικών στον διπλό φασικό χώρο.

Πρόκειται για καινοτόμα προσπάθεια βαθέματος στην κατανόηση της αναπαράστασης φασικού χώρου της ημικλασικής χρονικής εξέλιξης μικροσκοπικών φυσικών συστημάτων, στην κατανόηση εξισώσεων χρονικής εξέλιξης επαγόμενων από ψευδοδιαφορικούς τελεστές φασικού χώρου ή τελεστές Weyl φασικού χώρου. Η θεωρία που αναπτύσσεται μπορεί να πλαισιώσει την ανάπτυξη μεθόδων αριθμητικής επίλυσης σειράς προβλημάτων, σε αναπαράσταση φασικού χώρου, στην Ατομική Φυσική, στην Κβαντική Οπτική και στη Θεωρητική Χημεία, τα οποία είναι πλέον πειραματικά διερευνήσιμα.

Thursday 1/12/2022, 12:05-12:55

Room: E.324

Speaker: Rembert Duine (Eindhoven University of Technology)

Title: Antimagnons, the bosonic Klein paradox, and magnonic black holes

Abstract: The collective excitations of the magnetic order in the ground state of ferromagnets are called spin waves, or, quantum-mechanically, magnons. In this talk I will discuss how spin currents may be used to dynamically stabilize magnetic configurations at an energy maximum. The collective excitations of such configurations are negative energy excitations that are most conveniently described in terms of antimagnons. I will describe how the coupling of ordinary magnons with these antimagnons paves the way for schemes for magnon amplification via the bosonic Klein paradox, and for black-hole-horizon analogues for magnons.

Thursday 8/12/2022, 12:05-12:55

Room: E.324

Speaker: Stavros Komineas (University of Crete)

Title: Topological skyrmions and dynamics

Abstract: The Dzyaloshinskii-Moriya (DM) interaction breaks chiral symmetry in magnets and it is instrumental for the stabilisation of a periodic solution (termed the spiral) in one space dimension, and of topological solitons (termed skyrmions) in two space dimensions. In addition, the chiral DM interaction has a profound effect on the magnetization field dynamics. A skyrmion breathing mode (i.e., oscillations of the skyrmion radius) is due to the breaking of the conservation of the total magnetization by the DM interaction. We describe nonlinear breathing oscillations in an antiferromagnet (AFM). We show that an expansion of the skyrmion can lead to its collapsing into a singularity. The process can be efficient even when the skyrmion is only mildly excited.

Thursday 15/12/2022, 12:05-12:55

Room: E.324

Speaker: Jonatan Lenells (KTH Royal Institute of Technology)

Title: Boundary value problems for the Ernst equation

Abstract: For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the (elliptic) Ernst equation. By solving this equation with Dirichlet boundary conditions imposed along a disk, Neugebauer and Meinel in the 1990s derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. I will discuss some other boundary value problems for the Ernst equation that can also be solved exactly. The exact solutions are obtained by using the integrable structure of the Ernst equation. I will also discuss a boundary value problem for the hyperbolic version of the Ernst equation that is relevant for the collision of two plane gravitational waves.  

Tuesday 28/2/2023, 13:05-13:55

Room: E.324

Speaker: Alkis Tersenov (University of Crete)

Title: Καινούργιες a priori εκτιμήσεις στη μελέτη των μη γραμμικών εξισώσεων παραβολικού τύπου 

Tuesday 7/3/2023, 13:05-13:55

Room: E.324

Speaker: Spyridon Filippas (Université Paris-Saclay)

Title: Οn unique continuation for waves in singular media

Abstract: The question of unique continuation consists in asking whether a partial observation of a wave on a small set is sufficient to determine the whole wave. After presenting some fundamental results of the theory we will explain how one can obtain quantitative uniqueness results in singular media. The key ingredient will be a Carleman estimate, which will be combined with the recent techniques of Laurent-Létautaud who obtained similar results in a smooth context.

Teusday 21/3/2023, 13:05-13:55

Room: E.324

Speaker: Nikolaos Athanasiou (University of Crete)

Title: Formation of trapped surfaces in general relativity

Abstract: Trapped surfaces play an important role in general relativity as they typically signal the formation of singularities which are enclosed inside black holes. I will discuss their basic properties and give a historical overview of the trapped surface formation problem. Time-permitting, I will present recent progress concerning the Einstein-Yang-Mills system.

Tuesday 25/4/2023, 13:05-13:55

Room: E.324

Speaker: Olena Gomonay (Johanes Gutenberg University)

Title: Elastic manipulation of antiferromagnetic domain structure

Abstract: Since the recent prediction [1] and discovery [2] of the Neel spin-orbit torques, antiferromagnets became active elements of spintronic devices in which antiferromagnetic states are manipulated electrically. However, many of antiferromagnets that are interesting for applications show strong magnetoelastic coupling which cannot be ignored in description and interpretation of switching processes. On the other hand, this coupling can be used for effective and energy consuming manipulation of antiferromagnetic states via external stresses and strains.

In this presentation we discuss magnetoelastic origin of the domain structure in a tetragonal antiferromagnet and formulate general criteria which define formation of the particular domain shapes depending on the sample geometry and the external stress. By comparing domain structures in presence of the electrical and stress-induced torques we derive equivalence between both stimuli and demonstrate possibility of strain-induced switching. We illustrate our predictions with recent experimental observations which underline importance and efficiency of strain-induced switching in antiferromagnetic-based films.


[1] J. Železný, et al, Phys. Rev. Lett. 113, 157201 (2014).

[2] P. Wadley, et al, Science 351 (2016).

Tuesday 9/5/2023, 13:05-13:55

Room: E.324

Speaker: Warren Li (Princeton University)

Title: The structure of singularities for spherically symmetric solutions of Einstein's equations with scalar matter

Abstract: By the Hawking-Penrose singularity theorems, there is a large class of 'singular' (i.e. incomplete) spacetimes solving the Einstein field equations. However, these theorems do not provide a mechanism of how such 'singularities' are formed, or any description of the structure of spacetime near singularity.

In this talk, upon reducing to the spherically symmetric case and adding a scalar field, we provide a study of the types of singularity that arise for general initial data, including naked singularities, spacelike singularities and (weak) null singularities.