# Karlstad Applied Analysis Seminar (KAAS)

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## Future seminars:

** **

**Talk-1**

**When:** 02:15PM-03:00PM, 21 September 2023

**What: ** Boundary conditions for the Boltzmann equation derived from a kinetic model of gas-surface interaction

**Who:** Kazuo Aoki , Kyoto University, Japan and National Cheng Kung University,Taiwan.

**Where:**** ** **offline**:21E415A **online**: https://kau-se.zoom.us/j/63372480848

**Abstract:**Boundary conditions for the Boltzmann equation are investigated on the basis of a kinetic model for gas-surface interaction. The model takes into account gas and physisorbed molecules interacting with a surface potential and colliding with surface and bulk phonons. The interaction layer is assumed to be thinner than the mean free path of the gas molecules, and the phonons are assumed to be at equilibrium. The asymptotic kinetic equation for the inner physisorbate layer, which forms a steady half-space problem, is derived and used to investigate boundary conditions for the Boltzmann equation that is valid outside the physisorbate layer. To be more specific, new models of the boundary condition are proposed on the basis of iterative solutions of the half-space problem and are assessed by the direct numerical analysis of the problem. In addition, some rigorous mathematical results for the half-space problem are presented.

This is a joint work with Vincent Giovangigli (Ecole Polytechnique), François Golse (Ecole Polytechnique) and Shingo Kosuge (Kyoto University).

**Talk-2**

**When:** 10:30AM-11:15AM, 11 October 2023

**What: ** A new hybrid Boltzmann-BGK model: consistency, hydrodynamic limits and applications.

**Who:** Giorgio Martalò , University of Parma, Italy.

**Where:**** ** **offline**:21E415A **online**: https://kau-se.zoom.us/j/63372480848

**Abstract:**The evolution of a gas is classically described by the Boltzmann equation and the con- tribution of interactions is modeled by proper integral nonlinear operators. Unfortunately this approach requires a high computational cost in simulations for gas mixtures due to a larger number of collision operators (one for each type of interaction) [1].

For such reason, alternative formulations have been proposed since the pioneering model for a single gas proposed by Bhatnagar, Gross and Krook [2], whose idea was to replace the integral nonlinear operator by a simpler linear one reproducing the relaxation of the system towards a Maxwellian state and recovering the usual conservations properties. The extension of this approach to mixtures is not unique; a consistent BGK model has been pro- posed recently in [3], where the sum of Boltzmann terms is replaced by a sum of relaxation operators, one for each couple of components.

In this talk we want to propose a new mixed model combining the positive features of Boltzmann and BGK descriptions, preserving the accuracy of Boltzmann description for a part of the collisional phenomenon and using the BGK approach for the remaining processes. The consistency of the model is proved, focusing in particular on the preservation of global momentum and total energy, convergence to a global (Maxwellian) equilibrium; moreover, the existence of a Lyapunov functional miming the entropy dissipation is guaranteed.

This model is very useful to describe different hydrodynamic regimes, like the one dom- inated by intraspecies collisions, typical for mixture whose components have very disparate masses (e.g. ions and electrons) [4]. For such regime, we derive macroscopic equations of Euler and Navier-Stokes type and we test them on the classical shock wave problem.

This work is in collaboration with M. Bisi, M. Groppi, E. Lucchin and A. Macaluso (University of Parma).

[1] Kosuge S.: Model Boltzmann equation for gas mixtures: Construction and numerical comparison. Eur. J. Mech. B Fluids 28 n.1, 170–184 (2009)

[2] Bhatnagar P.L., Gross E.P., Krook M.: A model for collision processes in gases. Phys. Rev. 94 n.3, 511–524 (1954)

[3] Bobylev A.V., Bisi M., Groppi M., Spiga G., Potapenko I.F.: A general consistent BGK model for gas mixtures. Kinet. Relat. Models 11 n.6, 1377–1393 (2018)

[4] Galkin V.S., Makashev N.K.: Kinetic derivation of the gas-dynamic equation for multicomponent mix- tures of light and heavy particles. Fluid Dyn. 29 n.1, 140–155 (1994).

**Talk-3**

**When:** 10:30AM-11:30AM, 13 October 2023

**What: ** Mathematical models for epidemics (including Covid-19)

**Who:** Tom Britton, Stockholm University.

**Where:**** ** **offline**:9C203 **online**: https://kau-se.zoom.us/j/63372480848

**Abstract: **In the talk I will first describe the basics for mathematical epidemic models, and indicate the many extensions that exist. Then I will briefly describe three problems I (and many others) have worked on during Covid-19: Herd immunity, Optimizing interventions in time and magnitude, and the significance of the generation time distribution and why it is hard to infer.

**Talk-4**

**When:** 10:30AM-11:15AM, 22 November 2023

**What: ** Piecewise deterministic Monte Carlo.

**Who:** Joris Bierkens , Delft Institute of Applied Mathematics, TU Delft, The Netherlands.

**Where:**** ** https://kau-se.zoom.us/j/63372480848

**Abstract:**In recent years piecewise deterministic Markov processes (PDMPs) have emerged as a promising alternative to classical MCMC algorithms. In particular these PDMP based algorithms have good convergence properties and allow for efficient subsampling.

Many different PDMP based algorithms can be designed, but among the most fundamental ones are the Bouncy Particle sampler, the Zig-Zag sampler and the Boomerang Sampler.

In this talk these algorithms will be introduced and a comparison of properties of these algorithms will be presented, such as e.g. results on efficient subsampling, ergodicity and scaling with respect to dimension.

**Talk-5**

**When:** 10:30AM-11:15AM, 29 November 2023

**What: ** A gradient flow structure for nonlocal transport equations with nonlinear mobility.

**Who:** Oliver Tse , Eindhoven University of Technology, The Netherlands.

**Where:**** ** https://kau-se.zoom.us/j/63372480848

**Abstract:** Nonlocal transport equations with nonlinear mobility (NTNs) are a class of conservation laws that commonly arise as large population limits of complex phenomena manifested in models for biological processes with overcrowding prevention such as bacterial chemotaxis, the collective behavior of animal groups, and vehicular and pedestrian movements. Despite the abundance of gradient-flow techniques available for nonlocal transport equations, many of these tools are not directly applicable to NTNs. In this talk, I will highlight the main ideas and strategy involved in giving 1-d NTNs a rigorous gradient flow structure via the evolutionary Gamma-convergence of a family of discrete particle approximations.