Speaker: **Jan Blechta**

Title: **GMRES with I+compact is superlinear. What about A+compact?**

Abstract: It is known for decades that the Krylov subspace methods applied to compact perturbations of identity produce superlinear convergence and that the convergence behavior is bounded in terms of the distribution of the singular values of the perturbation. In this talk we will present a generalization of such kind of result. We consider operators that exhibit linear GMRES convergence. Such operators are precisely those that do not contain the origin in the closure of their numerical range. The main result is stability with respect to compact perturbations: If the operator is compactly perturbed, deviation from the linear convergence is controlled in terms of the singular values of the perturbation. We will give this statement a precise quantitative meaning, which allows for arbitrarily large perturbations. We will mention applications of this result and state some open questions. This result has been published as [J. Blechta, SIMAX 2021, https://doi.org/10. 1137/20M1340848].

Speaker: **Daniel Campbell**

Title:** Injectivity in second-gradient Nonlinear Elasticity**

Abstract: We study injectivity for models of Nonlinear Elasticity that involve the second gradient. We assume that Ω ⊂ R^n is a domain, f ∈ W^(2,q)(Ω, R^n) satisfies |J_f|^(−a) ∈ L^1 and that f equals a given homeomorphism on ∂Ω. Under suitable conditions on q and a we show that f must be a homeomorphism. As a main new tool we find an optimal condition for a and q that imply that H^(n−1)({J_f = 0}) = 0 and hence J_f cannot change sign. We further specify in dependence of q and a the maximal Hausdorff dimension d of the critical set {J_f = 0}. The sharpness of our conditions for d is demonstrated by constructing respective counterexamples.

]]>Speaker:** Lenka Slavíková**

Title: **Singular integral operators from different perspectives**

Abstract: In this talk, I will introduce my research, which involves (multi)-linear singular integral operators and their applications in ergodic theory, as well as somewhat related topics featuring the specific singular integral operator called fractional Laplacian and associated function spaces.

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Abstract: Planets and moons reorient in space due to mass redistribution that is associated with various types of internal and external processes. While the equilibrium orientation of a tidally locked body is well understood, much less explored are the dynamics of the reorientation process (or true polar wander, TPW, used here for the motion of either the rotation or the tidal pole). The talk will cover the basic theory and observations, and our numerical solution to the problem will be outlined.

]]>Speaker: **Eduard Feireisl**

Title: **Analysis of open fluid systems with random data**

Abstract: We discuss the Navier-Stokes-Fourier system describing the motion of a general compressible, viscous and heat conducting fluid, with general inhomogeneous random data. This includes the iconic examples of turbulent regimes: the Rayleigh-Benard convection problem, and the Taylor-Couette flow. Theoretical results are applied to problems of convergence of numerical schemes in particular the Monte Carlo and stochastic collocation methods.

]]>Speaker: **Miroslav Bulíček**

Title: **On uniqueness of the flow of incompressible fluids – from Navier–Stokes to Ladyzhenskaya and back to Euler**

Abstract: The question of the uniqueness of a (proper) solution to the equations describing the flow of incompressible fluids belongs to the most classical open problems in the analysis of PDE’s. The story started by the work of Leray 90 years ago. As usual during the decades there were many trends in the analysis of the Navier-Stokes and/or the Euler equations and their generalizations – the focus was on the regularity, partial regularity, uniqueness, conditional uniquenes/regularity, existence of measure valued solutions up to recent decade full of the results about the non-uniqueness of a solution implying that our concepts and understanding of the notion of the solution must be revised. We present a certain idea how to change the equations/constitutive laws only for extremely large shear-rates so that the modified Navier-Stokes or Euler equations have unique solution. Moreover, such a modification does not change the solution to the original problem at least on a short time interval. Hence, it can be understood as another concept of a solution with the key advantage – it exists globally in time and it is unique.

]]>Speaker: **Jan Vybíral**

Title: **A multivariate Riesz basis of ReLU neural networks**

Abstract: We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of L2([0,1]) based on the Gershgorin theorem. We also generalize this system to higher dimensions d>1 by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of L2([0,1]^d) can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of d, making it an attractive building block regarding future multivariate analysis of neural networks.

]]>Speaker: **Vít Průša** and **Karel Tůma**

Title:** A Thermodynamic Framework for Non-Isothermal Phenomenological Models for Mullins Effect**

Abstract: The Mullins effect is a common name for a family of intriguing inelastic responses of various solid materials, in particular filled rubbers. Given the importance of the Mullins effect, there have been many attempts to develop mathematical models describing the effect. However, most of available models focus exclusively on the mechanical response, and are restricted to the idealised isothermal setting. We lift the restriction to isothermal processes, and we propose a full thermodynamic framework for a class of phenomenological models of Mullins effect. In particular, we identify energy storage mechanisms (Helmholtz free energy) and entropy production mechanisms that on the level of stress–strain relation lead to the idealised Mullins effect or to the Mullins effect with permanent strain. The models constructed within the proposed framework can be used in modelling of fully coupled thermo-mechanical processes, and the models are guaranteed to be consistent with the laws of thermodynamics.

Next, the system of governing equations, that are derived in the Eulerian setting, are transformed to the Lagrangian setting and used for simulations of several different problems such as the investigation of the interplay between the Mullins effect and the Gough-Joule effect or the study of the Mullins effect for a cylinder subjected to combined extension and torsion. The model is implemented in FEniCS finite element.

Speaker: **Michiel Renger**

Title: **Macroscopic Fluctuation Theory on discrete spaces**

Abstract: Often thermodynamical phenomena are described microscopically by a randomly evolving particle system, or macroscopically by an evolution equation, and the two levels of description are connected by sending the number of particles to infinity. Onsager and Machlup postulated that microscopic systems in detailed balance (reversible Markov process) behave as a gradient flow on the macroscopic level. This principle is now well-understood and can be made precise via the theory of large deviations. In order to understand the behaviour of non-equilibrium systems (not in detailed balance/nonreversible), one commonly studies large deviations of particle densities and fluxes. Classically one can decompose the dynamics into a gradient flow component (dissipating free energy) and a Hamiltonian component (conserving energy). Such decomposition becomes more difficult on discrete spaces, and a choice needs to be made: either to decompose fluxes or to decompose forces into dissipative and conservative parts. Decomposing fluxes is consistent with the GENERIC formalism, which is only rarely applicable to dynamics on discrete spaces. Decomposing forces is more consistent with Macroscopic Fluctuation Theory, and yields a framework that is sufficiently general to apply to, for example zero-range processes, boundary-driven systems and chemical reactions.

This work lies on the boundary between probability, analysis and physics, but I will mostly focus on the analysis and physics part.

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