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Kinetic and Related Models

April 2021 , Volume 14 , Issue 2

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Captivity of the solution to the granular media equation
Julian Tugaut
2021, 14(2): 199-209 doi: 10.3934/krm.2021002 +[Abstract](925) +[HTML](180) +[PDF](309.18KB)

The goal of the current paper is to provide assumptions under which the limiting probability of the granular media equation is known when there are several stable states. Indeed, it has been proved in our previous works [17,18] that there is convergence. However, very few is known about the limiting probability, even with a small diffusion coefficient.

Diffusion limit of the Vlasov-Poisson-Boltzmann system
Hai-Liang Li, Tong Yang and Mingying Zhong
2021, 14(2): 211-255 doi: 10.3934/krm.2021003 +[Abstract](1069) +[HTML](197) +[PDF](497.32KB)

In the present paper, we study the diffusion limit of the classical solution to the unipolar Vlasov-Poisson-Boltzmann (VPB) system with initial data near a global Maxwellian. We prove the convergence and establish the convergence rate of the global strong solution to the unipolar VPB system towards the solution to an incompressible Navier-Stokes-Poisson-Fourier system based on the spectral analysis with precise estimation on the initial layer.

A general way to confined stationary Vlasov-Poisson plasma configurations
Yulia O. Belyaeva, Björn Gebhard and Alexander L. Skubachevskii
2021, 14(2): 257-282 doi: 10.3934/krm.2021004 +[Abstract](1029) +[HTML](181) +[PDF](463.25KB)

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain \begin{document}$ \Omega\subset\mathbb{R}^3 $\end{document} describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of \begin{document}$ \Omega $\end{document}. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

Mathematical modelling of collagen fibres rearrangement during the tendon healing process
José Antonio Carrillo, Martin Parisot and Zuzanna Szymańska
2021, 14(2): 283-301 doi: 10.3934/krm.2021005 +[Abstract](989) +[HTML](238) +[PDF](9790.55KB)

Tendon injuries present a clinical challenge to modern medicine as they heal slowly and rarely is there full restoration to healthy tendon structure and mechanical strength. Moreover, the process of healing is not fully elucidated. To improve understanding of tendon function and the healing process, we propose a new model of collagen fibres rearrangement during tendon healing. The model consists of an integro-differential equation describing the dynamics of collagen fibres distribution. We further reduce the model in a suitable asym-ptotic regime leading to a nonlinear non-local Fokker-Planck type equation for the spatial and orientation distribution of collagen fibre bundles. Due to its simplicity, the reduced model allows for possible parameter estimation based on data. We showcase some of the qualitative properties of this model simulating its long time asymptotic behaviour and the total time for tendon fibres to align in terms of the model parameters. A possible biological interpretation of the numerical experiments performed leads us to the working hypothesis of the importance of tendon cell size in patient recovery.

Shadow Lagrangian dynamics for superfluidity
Patrick Henning and Anders M. N. Niklasson
2021, 14(2): 303-321 doi: 10.3934/krm.2021006 +[Abstract](942) +[HTML](145) +[PDF](2408.64KB)

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds
Hyunjin Ahn, Seung-Yeal Ha and Woojoo Shim
2021, 14(2): 323-351 doi: 10.3934/krm.2021007 +[Abstract](1211) +[HTML](152) +[PDF](988.57KB)

We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [30] for a CS ensemble, and provide a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the model proposed in [30], our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for isothermal case, our model reduces to the previous CS model in [30] on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit \begin{document}$ d $\end{document}-sphere \begin{document}$ \mathbb{S}^d $\end{document}. We show that the asymptotic emergent dynamics of the proposed TCS model on the unit \begin{document}$ d $\end{document}-sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on \begin{document}$ \mathbb{S}^2 $\end{document}.

Projective integration schemes for hyperbolic moment equations
Julian Koellermeier and Giovanni Samaey
2021, 14(2): 353-387 doi: 10.3934/krm.2021008 +[Abstract](844) +[HTML](155) +[PDF](3325.43KB)

In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates
Prasanta Kumar Barik, Ankik Kumar Giri and Rajesh Kumar
2021, 14(2): 389-406 doi: 10.3934/krm.2021009 +[Abstract](1000) +[HTML](113) +[PDF](335.7KB)

In this article, the existence of mass-conserving solutions is investigated to the continuous coagulation and collisional breakage equation with singular collision kernels. Here, the probability distribution function attains singularity near the origin. The existence result is constructed by using both conservative and non-conservative truncations to the continuous coagulation and collisional breakage equation. The proof of the existence result relies on a classical weak \begin{document}$ L^1 $\end{document} compactness method.

2020 Impact Factor: 1.432
5 Year Impact Factor: 1.641
2020 CiteScore: 3.1




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