From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

Abstract

We study the Boltzmann equation on the d-dimensional torus in a perturbative setting around a global equilibrium under the Navier-Stokes linearisation. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a C 0 -semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently on the Knudsen number. Finally we show a Cauchy theory and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal and furthermore, this result only requires derivatives in the space variable and allows to connect to solutions to the incompressible Navier-Stokes equations in these spaces.Keywords: Boltzmann equation on the Torus; Incompressible Navier-Stokes hydrodynamical limit; Cauchy theory uniform in Knudsen number, exponential rate of convergence towards global equilibrium.

On the convergence of smooth solutions from Boltzmann to Navier–Stokes

“…The analysis of the latter has been developed in [Bri15,Guo06] by Guo and Briant in spaces of type H x,v . In the spirit of the work [GMM17], Briant, Merino-Aceituno and Mouhot in [BMAM19] have extended the range of validity of the theory to larger spaces with polynomial weights with no assumption on the derivatives in the velocity variable (note that by polynomial weights, we mean that in the linearization (1.2), the weight M 1/2 is replaced by the inverse of a polynomial function of type v q ).…”

“…More recently, Briant in [Bri15] and Briant, Merino-Aceituno and Mouhot in [BMAM19] obtained convergence to equilibrium results for the rescaled Boltzmann equation uniformly in the rescaling parameter using hypocoercivity and "enlargement methods", that enabled them to weaken the assumptions on the data down to Sobolev spaces with polynomial weights.…”

A spectral study of the linearized Boltzmann operator in $ L^2 $-spaces with polynomial and Gaussian weights

“…Both papers rely on the spectral study led by [13,33] of the inhomogeneous linearized Boltzmann operator in L 2 v H s x M −1/2 which dictates the asymptotic of S ε and ε −1 S ε Q. The theory of hydrodynamic limits for smooth solutions of the Boltzmann equation was partially extended to a larger class of Sobolev spaces with polynomial weights during the last decade: a Cauchy theory close to equilibrium was developped in [19] and their weak compacity with respect to ε was shown in [10]. The strong convergence could not be deduced as in [8,15] since the spectral decomposition from [13] was not known to hold in the case of polynomial weights.…”

“…In [3,10,19,23,25,32,31,4], the authors consider the linearized flow S Λ of some form of the Boltzmann equation near a steady state and show it admits in various weighted Sobolev spaces a splitting…”

“…However, there is no spectral gap on the whole space, so one can study the linearized operator in Fourier space with fixed spatial frequency ξ in order to construct a projector Π satisfying (22). Furthermore, the splitting (22) allows to develop a Cauchy theory near an equilibrium, but in the context of asymptotic analysis, that is to say when Λ and thus Σ(Λ), Π, σ depend on a small parameter ε, one also needs to know their behavior as ε goes to zero (see for instance [2,8,9,10]). This is why we perform a precise study of the spectrum for |ξ| 1, and the diffusive limit of the Boltzmann equation requires a second order expansion of the eigenvalues.…”