Incompressible navier‐stokes and euler limits of the boltzmann equation

Abstract

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( -[ ueu(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.It is now possible to make various formal expansions of f, in powers of E about some local or global Maxwellian. In particular the well-known Hilbert-Chapman-Enskog method [l] leads to a sequence of equations for the hydrodynamical variables corresponding, respectively, to the (non-dissipative) Euler equations, the Navier-Stokes equations, etc. From now on we always consider a collision operator Q corresponding, a la Grad [2], to cut-off hard potentials. An important step in the mathematical development of the theory was taken by Caflisch [7] (and by Caflisch and Papanicolau [8] for a discrete velocities model) who proved the following: let p, u, T be smooth solutions of Euler equations in a torus for r E [0, to]. Then there exists a solution f E ( r , T ; u ) of the

Fluid Dynamic Limits of the Kinetic Theory of Gases

“…In the same regime, short time convergence was obtained by A. DeMasi-R. Esposito-J. Lebowitz [23] by an argument similar to Caflisch's for the compressible limit, i.e. by means of a truncated Hilbert expansion.…”

From the Kinetic Theory of Gases to Continuum Mechanics

“…In general, truncated Hilbert expansions are not everywhere nonnegative, and are not exact solutions of the Boltzmann equation. One obtains exact solutions of the Boltzmann equation by adding to the truncated Hilbert expansion some appropriate remainder term, satisfying a variant of the Boltzmann equation that becomes weakly nonlinear for small enough ε (see for instance [10,11,2].) The truncated Hilbert expansion with the remainder term so constructed is a rigorous, pointwise asymptotic expansion (meaning that ε −n |F ε − ( f 0 + ε f 1 + .…”

“…The Navier-Stokes limit of the Boltzmann equation had also been established on finite time intervals by adapting the Caflisch method based on Hilbert truncated expansions, by DeMasi, Esposito and Lebowitz [11].…”

Asymptotic Analysis of a Slightly Rarefied Gas with Nonlocal Boundary Conditions

“…We remark that the velocity distribution function φ S1 at the order of ε is Maxwellian and that the functions ω S1 , u iS1 and τ S1 determining the Maxwellian φ S1 are governed by the incompressible Navier-Stokes equations (see e.g. [4,11]). …”