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The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials. (English) Zbl 1178.35290

This work is concerned with the derivation of the Navier-Stokes equations for incompressible viscous fluids in three space dimension from the well-known Boltzmann equation that governs the kinetic theory of rarefied, monatomic gases. That equation determines the molecular number density depending on time, position and velocity as a solution of an integro-differential equation whose integral part is provided by the collision integral. The authors assume that the collision integral satisfies certain inequalities. They consider a perturbed Maxwellian distribution depending on a divergence-free velocity field as an initial value for the Boltzmann equation to investigate the properties of the renormalized number density with a small perturbation parameter and the velocity field it generates. By employing a sophisticated asymptotic analysis involving several properties of Lebesgue spaces, they prove that this velocity field is asymptotically equivalent to the Leray solution of the incompressible Navier-Stokes equations. The mathematical structure of solutions are studied in great detail.

MSC:

35Q30 Navier-Stokes equations
82C40 Kinetic theory of gases in time-dependent statistical mechanics
35C20 Asymptotic expansions of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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