arXiv subjects

Marlies Pirner

Publications and source records attributed to Marlies Pirner.

At least 19 recordsLinked to original sources

arXiv2026

Long time behavior of Fokker-Planck equations for bosons and fermions

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor $f(1\pm f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the inclusion principle of bosons and the exclusion principle of fermions. Assuming that global solutions exist, we prove exponential decay of the solutions to the global equilibrium in a weighted $L^2$-space without a close-to-equilibrium assumption. Our analysis is in the spirit of an $L^2$-hypocoercivity method. Our main Lyapunov functional is constructed from a logarithmic relative entropy and the (nonlinear) projection of the solution to the manifold of local-in-$x$ equilibria.

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arXiv2026

BGK model for rarefied gas in a bounded domain

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the $L^\infty$ space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an $L^2$ coercive estimate. Next, we derive the a priori estimate by obtaining an $L^\infty$ bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the $L^\infty$ stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.

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arXiv2025

Numerical schemes for a multi-species quantum BGK model

This work is devoted to the numerical implementation of the quantum Bhatnagar- Gross-Krook (BGK) model for gas mixtures consisting of classical and quantum particles (fermions, bosons). We consider the model proposed by Bae, Klingenberg, Pirner, and Yun in 2021 and implement an Implicit-Explicit (IMEX) scheme due to the stiffness of the collision operator. A major obstacle is updating the parameters of quantum local equilibrium, which requires computing by inverting the relation between density and energy at every grid point in space and time. We address this difficulty by using the Lagrange multiplier method to minimize a potential function subject to constraints defined by specific moment equalities. Moreover, we analyze the convergence of mean velocity and temperature between the species both analytically and numerically. When a quantum component is included, we observe that the converging quantity is physical temperature, not the kinetic temperature. This differs from the mixture of classical species.

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arXiv2024

A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules

We consider a multi component gas mixture with translational and internal energy degrees of freedom without chemical reactions assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modeled by a system of kinetic Fokker-Planck equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperatures, H-theorem and we characterize the equilibrium as two Maxwell distributions where all temperatures coincide.

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arXiv2024

Hypocoercivity for the non-linear semiconductor Boltzmann equation

A kinetic model for semiconductor devices is considered on a flat torus. We prove exponential decay to equilibrium for this non-linear kinetic model by hypocoercivity estimates. This seems to be the first hypocoercivity result for this nonlinear kinetic equation for semiconductor devices without smallness assumptions. The analysis benefits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

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arXiv2024

A consistent kinetic Fokker-Planck model for gas mixtures

We propose a general multi-species Fokker-Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations from the kinetic two-species BGK model and compute explicitly the exchange terms of momentum and energy.

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arXiv2023

Hypocoercivity of the linearized BGK-equation with stochastic coefficients

In this paper we study the effect of randomness on a linearized BGK-model in one dimension. We prove exponential decay rate to a global equilibrium. This decay rate can be proven to be independent of the stochastic influence in a physical reasonable norm. We will further discuss the decay rate of the $n$-th derivative with respect to the stochastic variable of the solutions. Our strategy is based on Lyapunov's method. The matrices we need for a Lyapunov's estimate now depend on the stochastic variable. This requires a careful analysis of the random effect.

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arXiv2023

Linear Landau damping for a two-species Vlasov-Poisson system for electrons and ions

This paper concerns the linear Landau damping for the two species Vlasov-Poisson system for ions and electrons near Penrose stable equilibria. The result is an extension of the result on the one species Vlasov-Poisson equation by Mouhout and Villani. Different from their work we do not describe the ions as a background species but as a species which is also described by a separate Vlasov equation. We show an exponential decay of the electric energy for the linearised system near Penrose stable equilibria.

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arXiv2022

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

We consider a kinetic description of a multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure \cite{arXiv:2101.09047}. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

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arXiv2022

Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model

A reaction-kinetic model for a two-species gas mixture undergoing pair generation and recombination reactions is considered on a flat torus. For dominant scattering with a non-moving constant-temperature background the macroscopic limit to a reaction-diffusion system is carried out. Exponential decay to equilibrium is proven for the kinetic model by hypocoercivity estimates. This seems to be the first rigorous derivation of a nonlinear reaction-diffusion system from a kinetic model as well as the first hypocoercivity result for a nonlinear kinetic problem without smallness assumptions. The analysis profits from uniform bounds of the solution in terms of the equilibrium velocity distribution.

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arXiv2022

A review on a general multi-species BGK model: modelling, theory and numerics

In this article we focus on kinetic equations for gas mixtures since in applications one often has to deal with mixtures instead of a single gas. In particular we consider an approximation of the Boltzmann equation, the Bathnagar-Gross-Krook (BGK) equation. This equation is used in many applications because it is very efficient in numerical simulations. In this article, we recall a general BGK equation for gas mixtures which has free parameters. Specific choices of these free parameters lead to special cases in the literature. For this model, we provide an overview concerning modelling, theoretical results and numerics.

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arXiv2022

BGK model for multi-component gases near a global Maxwellian

In this paper, we establish the existence of the unique global-in-time classical solutions to the multi-component BGK model suggested in \cite{mixmodel} when the initial data is a small perturbation of global equilibrium. For this, we carefully analyze the dissipative nature of the linearized multi-component relaxation operator, and observe that the partial dissipation from the intra-species and the inter-species linearized relaxation operators are combined in a complementary manner to give rise to the desired dissipation estimate of the model We also observe that the convergence rate of the distribution function increases as the momentum-energy interchange rate between the different components of the gas increases.

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arXiv2021

A review on BGK models for gas mixtures of mono and polyatomic molecules

We consider the socalled Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of mixture a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom like rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.

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arXiv2021

Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law. For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.

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arXiv2021

A consistent BGK model with velocity-dependent collision frequency for gas mixtures

We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.

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arXiv2020

BGK model of the multi-species Uehling-Uhlenbeck equation

We propose a BGK model of the quantum Boltzmann equation for gas mixtures. We also provide a sufficient condition that guarantees the existence of equilibrium coefficients so that the model shares the same conservation laws and $H$-theorem with the quantum Boltzmann equation. Unlike the classical BGK for gas mixtures, the equilibrium coefficinets of the local equilibiriums for quantum multi-species gases are defined through highly nonlinear relations that are not explicitly solvable. We verify in a unified way that such nonlinear relations uniquely determine the equilibrium coefficients under the condition, leading to the well-definedness of our model.

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arXiv2020

Hypocoercivity for a BGK model for gas mixtures

We consider a kinetic model for a two component gas mixture without chemical reactions. Our goal is to study hypocoercivity for the linearized BGK model for gas mixtures in continuous phase space. By constructing an entropy functional, we can prove exponential relaxation to equilibrium with explicit rates. Our strategy is based on the entropy and spectral methods adapting Lyapunov's direct method as presented in [Achleitner-Arnold-Carlen, 2016] for the one species linearized BGK model. In comparison to the one species case, we start with two partial differential equations, one for each species. These equations are coupled due to interspecies interactions, which requires additional estimates on these interspecies terms.

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arXiv2018

Existence and uniqueness of mild solutions for BGK models for gas mixtures of polyatomic molecules

We consider two models for a two component gas mixture with translational and internal energy degrees of freedom described by a BGK approximation assuming that the number of particles of each species remains constant. The two species are allowed to have different degrees of freedom in internal energy and are modelled by a system of kinetic BGK equations featuring two interaction terms to account for momentum and energy transfer between the species. We consider two models in the literature and prove that the models satisfy a different equation of state than the ideal gas law. Moreover, we prove that for these models there exist a unique positive solution.

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