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This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation D( f) by weighted relative Fisher information of f with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at-most linearly growing in-time rate. As an application of these estimates, we show the convergence of any ( H- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher L -moment. More precisely, if the initial data have some (large enough) polynomial L -moment, then we obtain an algebraic decay. If the initial data have a stretched exponential L -moment, then we recover a stretched exponential decay. [ABSTRACT FROM AUTHOR]

This paper is devoted to the establishment at the formal level of a scaling leading from gas-particles (also called Eulerian-Lagrangian) models for monodisperse thick sprays towards multiphase flows models (also called Eulerian-Eulerian) including the volume fraction as an unknown. This passage to the limit involves the study of an inelastic collision operator acting on internal energies as well as velocities. [ABSTRACT FROM AUTHOR]

We propose a model based on the coupling of two Boltzmann-like equations for the study of the evolution of dust particles in a rarefied atmosphere, such as it can be found in the context of safety studies for the ITER project of nuclear fusion. When the typical size of a dust speck becomes too large, the numerical simulation of the system under study becomes too expensive, and one needs to introduce an asymptotic model in which the mass ratio between molecules and dust specks tends to 0. This model is constituted of a coupling (by a drag force term) between a Boltzmann equation and a Vlasov equation. A rigorous proof of the passage to the limit is given in the spatially homogeneous setting. It includes a new variant of Povzner’s inequality in which the vanishing mass ratio is taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions. [ABSTRACT FROM AUTHOR]

In this work, we consider the smoothness of the solutions to the full Landau equation. In particular, we prove that any classical solutions (such as the ones obtained by Guo in a “close to equilibrium” setting) become immediately smooth with respect to all variables. This shows that the Landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation. [ABSTRACT FROM AUTHOR]

We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6 π ν j where j is the current density of the particles, and of a friction term 6 π ν ρ u where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/ N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure $\frac{1}{N}\sum_{1\le k\le N}\delta_{x_{k},v_{k}}$ , where x k is the center of the k-th ball and v k its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, []] who considered the case of periodically distributed x k s with v k =0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays. [ABSTRACT FROM AUTHOR]