This paper presents a general methodology to design macroscopic fluid models that take into account localized kinetic upscaling effects. The fluid models are solved in the whole domain together with a localized kinetic upscaling that corrects the fluid model wherever it is necessary. This upscaling is obtained by solving a kinetic equation on the non-equilibrium part of the distribution functio...

We consider a vector kinetic BGK equation leading to isentropic gas dynamics in the relaxation limit. We prove the existence of a weak solution satisfying a kinetic entropy inequality for any initial data with nite energy. Key-words: isentropic gas dynamics { kinetic BGK model { kinetic entropy Mathematics Subject Classi cation: 35L65, 76N15, 35B35, 82C40 Work partially supported by European TM...

We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit (t, x) → (t/ε, x/ε). We derive a new type of limiting Hamilton-Jacobi equation, which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity. We prove well-posedness of the phase problem and convergence towards the viscosity sol...

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under conideration include the kinetic Cucker-Smale equation [6, 7] with possibly non-symmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [14]. The main tools employed in the analysis are the velocity...

The model of inelastic Maxwell particles (IMP) allows one to derive some exact results which show the strong influence of inelasticity on the nonequilibrium properties of a granular gas. The aim of this work is to propose a simple model kinetic equation that preserves the most relevant properties of the Boltzmann equation (BE) for IMP and reduces to the BGK kinetic model in the elastic limit. I...

A kinetic model for the Boltzmann equation is proposed and explored as a practical means to investigate the properties of a dilute granular gas. It is shown that all spatially homogeneous initial distributions approach a universal "homogeneous cooling solution" after a few collisions. The homogeneous cooling solution (HCS) is studied in some detail and the exact solution is compared with known ...

The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear dispersive system. A direct application of the standard kinetic equations would miss interesting dynamics of the energy transfer process starting from a large-scale excitation. This failure is traced to an enforcement of the exact resonance conditi...

A model kinetic equation is proposed to describe the time evolution of a gas composed of particles which collide inelastically. Dissipation in collisions is described by means of a parameter which is related to the coefficient of restitution, e . The kinetic equation can be solved exactly for the homogeneous cooling state, providing explicit expressions for both the time-dependent ‘‘temperature...