# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

August 2019 , Volume 12 , Issue 4

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2019, 12(4): 681-701 doi: 10.3934/krm.2019026 +[Abstract](644) +[HTML](80) +[PDF](393.75KB)
Abstract:

In this paper, we are interested in the collective friction of a cloud of particles on the viscous incompressible fluid in which they are moving. The particle velocities are assumed to be given and the fluid is assumed to be driven by the stationary Stokes equations. We consider the limit where the number \begin{document}$N$\end{document} of particles goes to infinity with their diameters of order \begin{document}$1/N$\end{document} and their mutual distances of order \begin{document}$1/ N^{1/3}$\end{document}. The rigorous convergence of the fluid velocity to a limit which is solution to a stationary Stokes equation set in the full space but with an extra term, referred to as the Brinkman force, was proven in [5] when the particles are identical spheres in prescribed translations. Our result here is an extension to particles of arbitrary shapes in prescribed translations and rotations. The limit Stokes-Brinkman system involves the particle distribution in position, velocity and shape, through the so-called Stokes' resistance matrices.

2019, 12(4): 703-726 doi: 10.3934/krm.2019027 +[Abstract](649) +[HTML](76) +[PDF](3140.28KB)
Abstract:

We present a numerical algorithm for evaluating the Boltzmann collision operator with \begin{document}$O(N^2)$\end{document} operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Split and non-split forms of the collision operator are considered, which are forms of the collision operator that have separate and simultaneous evaluations of the gain and loss terms, respectively. Smaller numerical errors are observed in the conserved quantities in simulations using the non-split form.

2019, 12(4): 727-748 doi: 10.3934/krm.2019028 +[Abstract](762) +[HTML](86) +[PDF](388.48KB)
Abstract:

This paper is devoted to the diffusion and anomalous diffusion limit of the Fokker-Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We use probabilistic methods to recover and extend the results obtained in [22]. We prove in particular, in the critical case where the classical diffusion coefficient is no more defined, that the small mean free path limit gives rise to a diffusion equation, with an anomalous time scaling and with a variance breaking.

2019, 12(4): 749-764 doi: 10.3934/krm.2019029 +[Abstract](722) +[HTML](90) +[PDF](337.8KB)
Abstract:

In this paper, we are concerned with the boundary value problem in a slab for the stationary relativistic BGK model of Marle type, which is a relaxation model of the relativistic Boltzmann equation. In the case of fixed inflow boundary conditions, we establish the existence of unique stationary solutions.

2019, 12(4): 765-789 doi: 10.3934/krm.2019030 +[Abstract](699) +[HTML](73) +[PDF](424.78KB)
Abstract:

We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is \begin{document}$T_0$\end{document}, and the gas is initially in equilibrium with density \begin{document}$\rho_0$\end{document} and temperature \begin{document}$T_0$\end{document}. The temperature of the wall is suddenly changed to \begin{document}$T_w$\end{document} at time \begin{document}$t = 0$\end{document} and is kept at \begin{document}$T_w$\end{document} afterward. We study the quantitative short time behavior of the gas in response to the abrupt change of the wall temperature on the basis of the linearized Boltzmann equation. Our approach is based on a straightforward calculation of the exact formulas derived by Duhamel's integral. Our method allows us to establish the pointwise estimates of the microscopic distribution and the macroscopic variables in short time. We show that the short-time solution consists of the free molecular flow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary.

2019, 12(4): 791-827 doi: 10.3934/krm.2019031 +[Abstract](632) +[HTML](80) +[PDF](534.93KB)
Abstract:

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density ft, in the single particle phase-space, of a collection of interacting particles confined to move on the one-dimensional torus. The corresponding stochastic differential equation for the position and velocity of the particles is a conditional McKean-Vlasov type of evolution (conditional in the sense that the process depends on its own law through its own conditional expectation). In this paper, we study existence and uniqueness of the solution of the PDE in consideration. Challenges arise from the fact that the PDE is neither elliptic (the linear part is only hypoelliptic) nor in gradient form. Moreover, for some specific choices of the interaction function and for the simplified case in which the density profile does not depend on the spatial variable, we show that the model exhibits multiple stationary states (corresponding to the particles forming a coordinated clockwise/anticlockwise rotational motion) and we study convergence to such states as well. Finally, we prove mean-field convergence of an appropriate N-particles system to the solution of our PDE: more precisely, we show that the empirical measures of such a particle system converge weakly, as $N \to \infty$, to the solution of the PDE.

2019, 12(4): 829-884 doi: 10.3934/krm.2019032 +[Abstract](808) +[HTML](83) +[PDF](569.2KB)
Abstract:

We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.

2019, 12(4): 885-908 doi: 10.3934/krm.2019033 +[Abstract](759) +[HTML](80) +[PDF](2502.62KB)
Abstract:

As an extension of our previous work in [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in [43] in many scenarios, hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.

2019, 12(4): 909-922 doi: 10.3934/krm.2019034 +[Abstract](689) +[HTML](76) +[PDF](330.42KB)
Abstract:

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.

2019, 12(4): 923-944 doi: 10.3934/krm.2019035 +[Abstract](738) +[HTML](76) +[PDF](366.19KB)
Abstract:

This paper is concerned with the rarefaction waves for a model system of hyperbolic balance laws in the whole space and in the half space. We prove the asymptotic stability of rarefaction waves under smallness assumptions on the initial perturbation and on the amplitude of the waves. The proof is based on the \begin{document}$L^2$\end{document} energy method.

2018  Impact Factor: 1.38