From the Boltzmann equation to the incompressible Navier-Stokes equations on the torus: a quantitative error estimate
The author is interested in the Boltzmann equation as the Knudsen number tends to 0. After some manipulations, this leads to the equation \[ \partial_th_{\epsilon}+\frac{1}{\epsilon}\nu\cdot\nabla_xh_{\epsilon}=\frac{1}{\epsilon^2}L(h_{\epsilon})+\frac{1}{\epsilon}\Gamma(h_{\epsilon},h\epsilon), \tag{f} \] where \(L(h)\) and \(\Gamma(g,h)\) are certain functions. The purpose of this paper is to study the existence of solutions and exponential decay for ({f}). Thus the limit Knudsen number tends to 0 corresponds to \(\epsilon\to 0\). The main results are {\parindent=6mm \begin{itemize} \item[-] A proof by induction on \(s\), the order of Sobolev spaces, for the norm to the solution of ({f}). \item [-] A proof that the solution of an equation similar to the linearized ({f}) satisfies a Sobolev space bound. \item [-] An exponential decay theorem for the previous equation, again proved by induction on \(s\). \item [-] A proof of strong convergences for some variables, which shows that these variables solve the incompressible Navier-Stokes and Boussinesq equation. \end{itemize}} The proofs use Gronvall, Cauchy-Schwarz and Poincaré type inequalities. In an appendix, the assumptions are validated.
incompressible Navier-Stokes hydrodynamical limit
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