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Kinetic & Related Models

June 2020 , Volume 13 , Issue 3

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High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials
Zheng-an Yao and Yu-Long Zhou
2020, 13(3): 435-478 doi: 10.3934/krm.2020015 +[Abstract](222) +[HTML](94) +[PDF](540.44KB)

The Boltzmann equation has an angular singularity inherent in the long range interaction between molecules. Angular singularity produces many difficulties in both theoretical and numerical study of Boltzmann equation. As a result, many rely on angular cutoff models to approximate the Boltzmann equation. Cutoff models have no angular singularity and can be solved by existent numerical methods. However, as the singularity goes stronger, the proportion of grazing collision becomes larger, which renders ineffectiveness of the cutoff Boltzmann equation. Based on the theoretical result that the limit of grazing collision is Landau operator, we propose to add a suitably scaled Landau operator to the cutoff equation to form a new approximate equation. This new approximate equation was studied in [27] in the case of hard potentials and its approximation accuracy is proved to be one order higher than that of angular cutoff models, which is a significant improvement in numerical computing. In this work, under moderately soft potentials, we establish the well-posedness theory of the new approximate equation, prove regularity propagation of its solution, check the high order accuracy. The new approximate equation can be solved by existing numerical methods, and this work may provide a theoretical foundation and a new direction to high order numerical methods for solving the Boltzmann equation.

A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators
Pierre Patie and Aditya Vaidyanathan
2020, 13(3): 479-506 doi: 10.3934/krm.2020016 +[Abstract](193) +[HTML](79) +[PDF](661.85KB)

The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup \begin{document}$ \widetilde{{P}} = (e^{-t\widetilde{{\mathbf{A}}}})_{t \geqslant 0} $\end{document} to a target semigroup \begin{document}$ P $\end{document} which is the object of study. This allows us to obtain conditions under which \begin{document}$ P $\end{document} satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of \begin{document}$ \widetilde{{\mathbf{A}}} $\end{document}. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on \begin{document}$ \mathbb{R}^d $\end{document}, and non-local Jacobi semigroups on \begin{document}$ [0,1]^d $\end{document}, which have been introduced and studied for \begin{document}$ d = 1 $\end{document} in [12]. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants.

Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons
José A. Carrillo, Katharina Hopf and Marie-Therese Wolfram
2020, 13(3): 507-529 doi: 10.3934/krm.2020017 +[Abstract](223) +[HTML](87) +[PDF](1999.05KB)

Kaniadakis and Quarati (1994) proposed a Fokker–Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily long time, and thus enabling a numerical study of the condensation process in the Kaniadakis–Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis–Quarati model in \begin{document}$ 3 $\end{document}D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

Asymptotic behavior for the Vlasov-Poisson equations with strong uniform magnetic field and general initial conditions
Mihaï Bostan
2020, 13(3): 531-548 doi: 10.3934/krm.2020018 +[Abstract](213) +[HTML](82) +[PDF](399.77KB)

We investigate the Vlasov-Poisson equations perturbed by a strong external uniform magnetic field. We study the asymptotic behavior of the solutions, based on averaging techniques. We analyze the case of general initial conditions. By filtering out the oscillations, we are led to a profile. We prove strong convergence results and establish second order estimates.

Cercignani-Lampis boundary in the Boltzmann theory
Hongxu Chen
2020, 13(3): 549-597 doi: 10.3934/krm.2020019 +[Abstract](230) +[HTML](74) +[PDF](845.96KB)

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an \begin{document}$ L^\infty $\end{document} estimate. In particular, for the \begin{document}$ L^\infty $\end{document} bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Moreover, under some constraints on the wall temperature and the accommodation coefficients, we construct a unique steady solution of the Boltzmann equation.

The Vlasov-Maxwell-Boltzmann system near Maxwellians with strong background magnetic field
Yuanjie Lei and Huijiang Zhao
2020, 13(3): 599-621 doi: 10.3934/krm.2020020 +[Abstract](185) +[HTML](70) +[PDF](452.52KB)

In this paper, we are concerned with the construction of global-in-time solutions of the Cauchy problem of the Vlasov-Maxwell-Boltzmann system near Maxwellians with strong uniform background magnetic field. The background magnetic field under our consideration can be any given non-zero constant vector rather than vacuum in the previous results available up to now. Our analysis is motivated by the nonlinear energy method developed recently in [16,24,25] for the Boltzmann equation and the key point in our analysis is to deduce the dissipation estimates of the electronic field and strong background magnetic field.

Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes equations in the whole domain
Jeongho Kim and Weiyuan Zou
2020, 13(3): 623-651 doi: 10.3934/krm.2020021 +[Abstract](178) +[HTML](85) +[PDF](482.48KB)

We study local existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker–Smale (in short TCS) model coupled with incompressible Navier–Stokes (NS) equations in the whole space. The coupled system consists of the kinetic TCS equation for particle ensemble and the incompressible NS equations for a fluid via a drag force. For the strong solution, we investigate the blow-up mechanism for the coupled system, and we also study the global existence of a weak solution in the whole space.

2018  Impact Factor: 1.38




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