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Kinetic & Related Models

October 2019 , Volume 12 , Issue 5

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Macroscopic regularity for the relativistic Boltzmann equation with initial singularities
Yan Yong and Weiyuan Zou
2019, 12(5): 945-967 doi: 10.3934/krm.2019036 +[Abstract](372) +[HTML](85) +[PDF](390.27KB)
Abstract:

In this paper, it is proved that the macroscopic parts of the relativistic Boltzmann equation will be continuous, even though the macroscopic components are discontinuity initially. The Lorentz transformation plays an important role to prove the continuity of nonlinear term.

Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs
Shi Jin and Yingda Li
2019, 12(5): 969-993 doi: 10.3934/krm.2019037 +[Abstract](393) +[HTML](89) +[PDF](434.84KB)
Abstract:

In this paper we study the general discrete-velocity models of Boltzmann equation with uncertainties from collision kernel and random inputs. We follow the framework of Kawashima and extend it to the case of diffusive scaling in a random setting. First, we provide a uniform regularity analysis in the random space with the help of a Lyapunov-type functional, and prove a uniformly (in the Knudsen number) exponential decay towards the global equilibrium, under certain smallness assumption on the random perturbation of the collision kernel, for suitably small initial data. Then we consider the generalized polynomial chaos based stochastic Galerkin approximation (gPC-SG) of the model, and prove the spectral convergence and the exponential time decay of the gPC-SG error uniformly in the Knudsen number.

Sedimentation of particles in Stokes flow
Amina Mecherbet
2019, 12(5): 995-1044 doi: 10.3934/krm.2019038 +[Abstract](319) +[HTML](94) +[PDF](513.56KB)
Abstract:

In this paper, we consider \begin{document}$ N $\end{document} identical spherical particles sedimenting in a uniform gravitational field. Particle rotation is included in the model while fluid and particle inertia are neglected. Using the method of reflections, we extend the investigation of [11] by discussing the threshold beyond which the minimal particle distance is conserved for a short time interval independent of \begin{document}$ N $\end{document}. We also prove that the particles interact with a singular interaction force given by the Oseen tensor and justify the mean field approximation in the spirit of [8] and [9].

A probabilistic approach for the mean-field limit to the Cucker-Smale model with a singular communication
Seung-Yeal Ha, Jeongho Kim, Peter Pickl and Xiongtao Zhang
2019, 12(5): 1045-1067 doi: 10.3934/krm.2019039 +[Abstract](369) +[HTML](110) +[PDF](393.22KB)
Abstract:

We present a probabilistic approach for derivation of the kinetic Cucker-Smale (C-S) equation from the particle C-S model with singular communication. For the system we are considering, it is impossible to validate effective description for certain special initial data, thus such a probabilistic approach is the best one can hope for. More precisely, we consider a system in which kinetic trajectories are deviated from a microscopic model and use a suitable probability measure to quantify the randomness in the initial data. We show that the set of "bad initial data" does in fact have small measure and that this small probability decays to zero algebraically, as \begin{document}$ N $\end{document} tends to infinity. For this, we introduce an appropriate cut-off in the communication weight. We also provide a relation between the order of the singularity and the order of the cut-off such that the machinery for deriving classical mean-field limits introduced in [3] can be applied to our setting.

The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation
Jacek Banasiak, Luke O. Joel and Sergey Shindin
2019, 12(5): 1069-1092 doi: 10.3934/krm.2019040 +[Abstract](292) +[HTML](81) +[PDF](1348.06KB)
Abstract:

In this paper we study the discrete coagulation–fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.

Differentiability in perturbation parameter of measure solutions to perturbed transport equation
Piotr Gwiazda, Sander C. Hille, Kamila Łyczek and Agnieszka Świerczewska-Gwiazda
2019, 12(5): 1093-1108 doi: 10.3934/krm.2019041 +[Abstract](274) +[HTML](98) +[PDF](368.57KB)
Abstract:

We consider a linear perturbation in the velocity field of the transport equation. We investigate solutions in the space of bounded Radon measures and show that they are differentiable with respect to the perturbation parameter in a proper Banach space, which is predual to the Hölder space \begin{document}$ \mathcal{C}^{1+\alpha}( {\mathbb{R}^d}) $\end{document}. This result on differentiability is necessary for application in optimal control theory, which we also discuss.

Kinetic methods for inverse problems
Michael Herty and Giuseppe Visconti
2019, 12(5): 1109-1130 doi: 10.3934/krm.2019042 +[Abstract](373) +[HTML](87) +[PDF](1660.59KB)
Abstract:

The Ensemble Kalman Filter method can be used as an iterative numerical scheme for parameter identification ornonlinear filtering problems. We study the limit of infinitely large ensemble size and derive the corresponding mean-field limit of the ensemble method. The solution of the inverse problem is provided by the expected value of the distribution of the ensembles and the kinetic equation allows, in simple cases, to analyze stability of these solutions. Further, we present a slight but stable modification of the method which leads to a Fokker-Planck-type kinetic equation. The kinetic methods proposed here are able to solve the problem with a reduced computational complexity in the limit of a large ensemble size. We illustrate the properties and the ability of the kinetic model to provide solution to inverse problems by using examples from the literature.

On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks
Bin Li
2019, 12(5): 1131-1162 doi: 10.3934/krm.2019043 +[Abstract](319) +[HTML](105) +[PDF](416.87KB)
Abstract:

In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.

Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations
Ricardo J. Alonso, Véronique Bagland and Bertrand Lods
2019, 12(5): 1163-1183 doi: 10.3934/krm.2019044 +[Abstract](266) +[HTML](89) +[PDF](383.57KB)
Abstract:

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in [24,26,15,23].

Diffusion limit for a kinetic equation with a thermostatted interface
Giada Basile, Tomasz Komorowski and Stefano Olla
2019, 12(5): 1185-1196 doi: 10.3934/krm.2019045 +[Abstract](325) +[HTML](70) +[PDF](328.02KB)
Abstract:

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature \begin{document}$ T $\end{document} in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution \begin{document}$ \rho(t, y) $\end{document} of a heat equation with the boundary condition \begin{document}$ \rho(t, 0)\equiv T $\end{document}.

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