Tue, 21 May 2019
14:15
L4

A simple proof of the classification of unitary highest weight modules

Pavle Pandzic
(University of Zagreb)
Abstract

Unitary highest weight modules were classified in the 1980s by Enright-Howe-Wallach and independently by Jakobsen. The classification is based on a version of the Dirac inequality, but the proofs also require a number of other techniques and are quite involved. We present a much simpler proof based on a different version of the Dirac inequality. This is joint work with Vladimir Soucek and Vit Tucek.
 

Mon, 30 Oct 2017

16:00 - 17:00
L4

Effects of small boundary perturbation on the porous medium flow

Igor Pazanin
(University of Zagreb)
Abstract

It is well-known that only a limited number of the fluid flow problems can be solved (or approximated) by the solutions in the explicit form. To derive such solutions, we usually need to start with (over)simplified mathematical models and consider ideal geometries on the flow domains with no distortions introduced. However, in practice, the boundary of the fluid domain can contain various small irregularities (rugosities, dents, etc.) being far from the ideal one. Such problems are challenging from the mathematical point of view and, in most cases, can be treated only numerically. The analytical treatments are rare because introducing the small parameter as the perturbation quantity in the domain boundary forces us to perform tedious change of variables. Having this in mind, our goal is to present recent analytical results on the effects of a slightly perturbed boundary on the fluid flow through a channel filled with a porous medium. We start from a rectangular domain and then perturb the upper part of its boundary by the product of the small parameter $\varepsilon$ and arbitrary smooth function. The porous medium flow is described by the Darcy-Brinkman model which can handle the presence of a boundary on which the no-slip condition for the velocity is imposed. Using asymptotic analysis with respect to $\varepsilon$, we formally derive the effective model in the form of the explicit formulae for the velocity and pressure. The obtained asymptotic approximation clearly shows the nonlocal effects of the small boundary perturbation. The error analysis is also conducted providing the order of accuracy of the asymptotic solution. We will also address the problem of the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude $\varepsilon$ is applied on the channel's walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number-regime is derived using asymptotic analysis with respect to $\varepsilon$. The obtained mathematical model clearly indicates the influence of the porous medium, chemical reaction and boundary distortion on the effective flow.

This is a joint work with Eduard Marušić-Paloka (University of Zagreb).

Mon, 10 Oct 2016

16:00 - 17:00
L4

Homogenization of thin structures in nonlinear elasticity - periodic and non-periodic

Igor Velcic
(University of Zagreb)
Abstract

We will give the results on the models of thin plates and rods in nonlinear elasticity by doing simultaneous homogenization and dimensional reduction. In the case of bending plate we are able to obtain the models only under periodicity assumption and assuming some special relation between the periodicity of the material and thickness of the body. In the von K\'arm\'an regime of rods and plates and in the bending regime of rods we are able to obtain the models in the general non-periodic setting. In this talk we will focus on the derivation of the rod model in the bending regime without any assumption on periodicity.

Thu, 29 Oct 2015

14:00 - 15:00
L4

Classifying $A_{\mathfrak{q}}(\lambda)$ modules by their Dirac cohomology

Pavle Pandzic
(University of Zagreb)
Abstract

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

Mon, 20 Oct 2014

17:00 - 18:00
L6

Asymptotic modelling of the fluid flow with a pressure-dependent viscosity

Igor Pazanin
(University of Zagreb)
Abstract
Our goal is to present recent results on the stationary motion of incompressible viscous fluid with a pressure-dependent viscosity. Under general assumptions on the viscosity-pressure relation (satisfied by the Barus formula and other empiric laws), first we discuss the existence and uniqueness of the solution of the corresponding boundary value problem. The main part of the talk is devoted to asymptotic analysis of such system in thin domains naturally appearing in the applications. We address the problems of fluid flow in pipe-like domains and also study the behavior of a lubricant flowing through a narrow gap. In each setting we rigorously derive new asymptotic model describing the effective flow. The key idea is to conveniently transform the governing problem into the Stokes system with small nonlinear perturbation.
This is a joint work with Eduard Marusic-Paloka (University of Zagreb).
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