1989
DOI: 10.1002/cpa.3160420810
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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)~… Show more

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Cited by 201 publications
(132 citation statements)
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“…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.…”
Section: Incompressible Navier-stokes Limitmentioning
confidence: 54%
“…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.…”
Section: Incompressible Navier-stokes Limitmentioning
confidence: 54%
“…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 + .…”
Section: The Compressible Euler Limit and Hilbert's Expansionmentioning
confidence: 99%
“…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].…”
Section: Family Of Renormalized Solutions Of the Boltzmann Equation (mentioning
confidence: 99%
“…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]). …”
Section: Fluid-dynamic Type Equationsmentioning
confidence: 99%