ISSN:

1937-5093

eISSN:

1937-5077

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## Kinetic & Related Models

August 2019 , Volume 12 , Issue 4

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**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

*+*[Abstract](800)

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**Abstract:**

We present a numerical algorithm for evaluating the Boltzmann collision operator with

*+*[Abstract](904)

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**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 [

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**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.

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**Abstract:**

We consider a semi-infinite expanse of a rarefied gas bounded by an infinite plane wall. The temperature of the wall is

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**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 *f _{t}*, 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.

*+*[Abstract](976)

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**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.

*+*[Abstract](942)

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**Abstract:**

As an extension of our previous work in [

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**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.

*+*[Abstract](1028)

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**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

2018 Impact Factor: 1.38

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