Weekly research seminar

2023-11-30, 12:30, 5070

Michał Fabisiak (doktorant SDNŚiS)

**Cucker-Smale model in bounded domains**Cucker-Smale model describes the behaviour of agents aligning their velocities according to nonlocal protocol. We consider the model posed in domains with a boundary and try to justify the mean-field limit between particle and kinetic levels. Due to the low regularity of kinetic solutions, we introd...

2023-11-23, 12:30, 5070

Mateusz Dembny (doktorant SDNŚiS)

**On differential Harnack bounds for a fractional heat equation**Consider the linear heat equation. The celebrated Li-Yau inequality states that for positive solutions we have $\bigtriangleup \log u \geq - \frac{n}{2t}$. By integrating this inequality along a straight space-time interval between two points, we may deduce the sharp Harnack estimate. In recent year...

2023-11-09, 12:30, 5070

Łukasz Chomienia (SDNŚiP)

**PDEs on low-dimensional structures: regularity and parabolic issues**By the low-dimensional structure we understand a closed subset of Euclidean space possessing some geometrical nature. The class includes, for instance, CW-complexes, families of manifolds, stratified manifolds etc. We very briefly recall the current state of the art of PDEs on such structures. ...

2023-10-26, 12:30, 5070

Jarosław Duda (Institute of Computer Science and Computer Mathematics, Jagiellonian University)

**Electromagnetism with built-in electric charge quantization as topological**I will discuss topological solitons starting with 1+1 dimensional sine-Gordon model. Then I will consider higher dimensional model, like topological defects with long-range e.g. Coulomb-like interactions observed in liquid crystals. To recreate electromagnetism for them (Faber's approach), we us...

2023-10-19, 12:30, 5070

Benoît Van Vaerenbergh (UCLouvain)

**The p-harmonic relaxation versus the Ginzburg-Landau functional**We will describe the manifold-valued harmonic extension problem of a boundary data defined on the boundary of a domain and taking values into the manifold. This extension has engineering applications, which we will present. Unfortunately, applying the direct method of...

2023-06-01, 12:30, 5070

Martin Ostoja-Starzewski (University of Illinois at Urbana-Champaign, USA)

See attached file ...

2023-05-25, 12:30, 5070

Benjamin Lledos (Université Paul Sabatier, Institut de Mathématiques de Toulouse)

**Some results about the uniqueness of the solutions in the calculus of variations**We want to find a framework in which we can establish the uniqueness of solutions for non-strictly convex problems in the calculus of variations. The main idea is to extend a method devised by Marcellini for a particular scenario. By examining various counterexamples and a simpler case, we demons...

2023-05-18, 12:30, 5070

Mateusz Dembny (SDNŚiS)

**Spiral vortex sheets and 2d Euler equation**In my talk, I will introduce Prandtl's and Kaden's spirals. Prandtl's spirals are weak solutions to the 2d Euler equation and this is a result by T. Cieslak, P. Kokocki and W.S. Ozanski. We will check whether Kaden's spirals are solutions to the 2d Euler equation. Also...

2023-05-04, 12:30, 5070

Michał Borowski (MIMUW)

**Absence of Lavrentiev’s phenomenon and Musielak-Orlicz-Sobolev spaces**We want to study Lavrentiev’s phenomenon for a broad class of variational functionals, covering anisotropic functionals of non-standard growth. To this purpose, we consider Musielak–Orlicz–Sobolev spaces and describe how the density of regular functions guarantees the absence of La...

2023-04-13, 12:30, 5070

Michał Fabisiak (doktorant SDNŚiS)

**Monokineticity and mean-field limit for strongly singular Cucker-Smale model**Cucker-Smale model, introduced in 2007, describes the evolution of particles alligning their velocities according to nonlocal interaction protocol. We will focus on the strongly singular case and see that, under some mild assumptions, measure valued solutions to kinetic Cucker-Smale equations are in...