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Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. (English) Zbl 0689.76024
Summary: We consider solutions of the Boltzmann equation in a d-dimensional torus, \(d=2,3\), (*) \(\partial_{\tau}f(r,v,\tau)+v\cdot \nabla_ rf=Q(f,f)\), for macroscopic times \(\tau =t/\epsilon^ N\), \(\epsilon \ll 1\), \(t\geq 0\), when the space vatiations are on a macroscopic scale \(x=\epsilon^{N-1}r\), \(N\geq 2\), x in the unit torus. Let u(x,t) be, for \(t\leq t_ 0\), 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 \(t\leq t_ 0\) which are close, to \(O(\epsilon^ 2)\) in a suitable norm, to the local Maxwellian [\({\bar \rho}/(2\pi\bar T)^{d/2}]\exp \{-[v-\epsilon u(x,t)]^ 2/2\bar T\}\) with contant density \({\bar \rho}\) and temperature \(\bar T.\) This is a particular case, defined by the choice of initialvalues of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For \(N\geq 3\) these equations correspond to variable density IE while for \(N^ 2\) they involve higher-order derivatives of the density.

MSC:
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
35Q30 Navier-Stokes equations
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