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.
References
[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.