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Kinetic & Related Models

December 2018 , Volume 11 , Issue 6

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Lagrangian solutions to the Vlasov-Poisson system with a point charge
Gianluca Crippa, Silvia Ligabue and Chiara Saffirio
2018, 11(6): 1277-1299 doi: 10.3934/krm.2018050 +[Abstract](58) +[HTML](16) +[PDF](594.04KB)

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [17], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [8] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.

Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces
Renjun Duan and Shota Sakamoto
2018, 11(6): 1301-1331 doi: 10.3934/krm.2018051 +[Abstract](47) +[HTML](31) +[PDF](533.17KB)

In this paper the Boltzmann equation near global Maxwellians is studied in the \begin{document} $d$ \end{document}-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable \begin{document} $(x,v)$ \end{document}. Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the \begin{document} $s$ \end{document}-order Besov space with \begin{document} $s≥ d/2$ \end{document} including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in \begin{document} $L^2$ \end{document} with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in \begin{document} $L^∞$ \end{document} framework through the technique of \begin{document} $L^2-L^∞$ \end{document} interplay. In contrast to hard potentials, \begin{document} $L^1$ \end{document} integrability in \begin{document} $x$ \end{document} of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches
Simone Göttlich, Stephan Knapp and Peter Schillen
2018, 11(6): 1333-1358 doi: 10.3934/krm.2018052 +[Abstract](46) +[HTML](21) +[PDF](652.68KB)

We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.

On the convergence to critical scaling profiles in submonolayer deposition models
Fernando P. da Costa, João T. Pinto and Rafael Sasportes
2018, 11(6): 1359-1376 doi: 10.3934/krm.2018053 +[Abstract](34) +[HTML](24) +[PDF](482.16KB)

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size \begin{document}$ n≥ 2$\end{document} for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction \begin{document}$ x = τ$\end{document} in the cluster size \begin{document}$ x$\end{document} vs. time \begin{document}$ \tau $\end{document} plane. In this paper we consider a different similarity variable, \begin{document}$ ξ : = (x-\tau )/\sqrt \tau $\end{document}, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile \begin{document}$ Φ_{2,n}(ξ)$\end{document} when \begin{document}$ x, \tau \to +∞$\end{document} with \begin{document}$ ξ$\end{document} fixed, as well as the rate at which the limit is approached.

A general consistent BGK model for gas mixtures
Alexander V. Bobylev, Marzia Bisi, Maria Groppi, Giampiero Spiga and Irina F. Potapenko
2018, 11(6): 1377-1393 doi: 10.3934/krm.2018054 +[Abstract](46) +[HTML](28) +[PDF](459.44KB)

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and H-theorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

Numerical study of an anisotropic Vlasov equation arising in plasma physics
Baptiste Fedele and Claudia Negulescu
2018, 11(6): 1395-1426 doi: 10.3934/krm.2018055 +[Abstract](36) +[HTML](24) +[PDF](1050.02KB)

Goal of this paper is to investigate several numerical schemes for the resolution of two anisotropic Vlasov equations. These two toy-models are obtained from a kinetic description of a tokamak plasma confined by strong magnetic fields. The simplicity of our toy-models permits to better understand the features of each scheme, in particular to investigate their asymptotic-preserving properties, in the aim to choose then the most adequate numerical scheme for upcoming, more realistic simulations.

Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus
Helge Dietert, Josephine Evans and Thomas Holding
2018, 11(6): 1427-1441 doi: 10.3934/krm.2018056 +[Abstract](42) +[HTML](24) +[PDF](369.2KB)

We study contraction for the kinetic Fokker-Planck operator on the torus. Solving the stochastic differential equation, we show contraction and therefore exponential convergence in the Monge-Kantorovich-Wasserstein \begin{document}$ \mathcal{W}_2$\end{document} distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.

Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles
Alexander Vibe and Nicole Marheineke
2018, 11(6): 1443-1474 doi: 10.3934/krm.2018057 +[Abstract](27) +[HTML](21) +[PDF](587.78KB)

In this paper we derive asymptotically the macroscopic bulk stress of a suspension of small inertial particles in an incompressible Newtonian fluid. We apply the general asymptotic framework to the special case of ellipsoidal particles and show the resulting modification due to inertia on the well-known particle-stresses based on the theory by Batchelor and Jeffery.

Linear Boltzmann dynamics in a strip with large reflective obstacles: Stationary state and residence time
Alessandro Ciallella and Emilio N. M. Cirillo
2018, 11(6): 1475-1501 doi: 10.3934/krm.2018058 +[Abstract](45) +[HTML](22) +[PDF](4732.66KB)

The presence of obstacles modifies the way in which particles diffuse. In cells, for instance, it is observed that, due to the presence of macromolecules playing the role of obstacles, the mean-square displacement of biomolecules scales as a power law with exponent smaller than one. On the other hand, different situations in grain and pedestrian dynamics in which the presence of an obstacle accelerates the dynamics are known. We focus on the time, called the residence time, needed by particles to cross a strip assuming that the dynamics inside the strip follows the linear Boltzmann dynamics. We find that the residence time is not monotonic with respect to the size and the location of the obstacles, since the obstacle can force those particles that eventually cross the strip to spend a smaller time in the strip itself. We focus on the case of a rectangular strip with two open sides and two reflective sides and we consider reflective obstacles into the strip. We prove that the stationary state of the linear Boltzmann dynamics, in the diffusive regime, converges to the solution of the Laplace equation with Dirichlet boundary conditions on the open sides and homogeneous Neumann boundary conditions on the other sides and on the obstacle boundaries.

Fractional diffusion limits of non-classical transport equations
Martin Frank and Weiran Sun
2018, 11(6): 1503-1526 doi: 10.3934/krm.2018059 +[Abstract](37) +[HTML](22) +[PDF](442.11KB)

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.

2016  Impact Factor: 1.261




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