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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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