Incompressible Navier - Stokes and Euler Limits of the Boltzmann Equation
نویسنده
چکیده
We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2,3, for macroscopic times T = t / e N , e a: 1, t 2 0, when the space variations are on a macroscopic scale x = eN-’r , N 2 2, x in the unit torus. Let u ( x , r ) be, for t to, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for r 6 to which are close, to O(e2) in a suitable norm, to the local Maxwellian [6/(2~T)~/’]exp( [ u eu(x, t)]’/2T} with constant density 6 and temperature 7. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N 2 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.
منابع مشابه
Fluid Dynamic Limits of the Kinetic Theory of Gases
These three lectures introduce the reader to recent progress on the hydrodynamic limits of the kinetic theory of gases. Lecture 1 outlines the main mathematical results in this direction, and explains in particular how the Euler or NavierStokes equations for compressible as well as incompressible fluids, can be derived from the Boltzmann equation. It also presents the notion of renormalized sol...
متن کاملFrom the Boltzmann equation to the Euler equations in the presence of boundaries
Abstract. The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012)...
متن کاملLattice Boltzmann Model for the Incompressible NavierStokes Equation
In the last decade or so, the lattice Boltzmann (LB) method has emerged as a new and effective numerical technique of computational fluid dynamics (CFD).(1-5) Modeling of the incompressible Navier-Stokes equation is among many of its wide applications. Indeed, the lattice Boltzmann equation (LBE) was first proposed to simulate the incompressible NavierStokes equations.(1) The incompressible Nav...
متن کاملThe Incompressible Navier – Stokes for the Nonlinear Discrete Velocity Models
We establish the incompressible Navier–Stokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge strongly to a limit governed ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006