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

Abstract

In this work, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. To justify this connection, our main idea is to use information on the limit system (for instance the fact that the Navier-Stokes equations are globally wellposed in two space dimensions or when the initial data is small). In particular we prove that the life span of the solutions to the rescaled Boltzmann equation is bounded from below by that of the Navier-Stokes system. We deal with general initial data in the whole space in dimensions 2 and 3, and also with well-prepared data in the case of periodic boundary conditions.

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

“…S ε (t) := exp(t(L − εv • ∇ x )), construct smooth solutions in L ∞ v H s x M −1/2 1 + |v| k with s > d/2, and prove their convergence as ε goes to zero. In [15], the authors prove a converse result in the same functional space ; as long as a solution f 0 to the incompressible Navier-Stokes-Fourier system exists, a solution f ε to (2) exists for ε small enough and f ε converges to f 0 . 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. This paper aims at providing such a generalization.…”

On the convergence from Boltzmann to Navier-Stokes-Fourier for general initial data

“…This is made possible by adapting the strategy from [7] of writing the solution to the Boltzmann equation as the sum a part with polynomial decay and a second one with Gaussian decay. The Gaussian part is treated with an approach reminiscent of the one from [17].…”

“…In [17], I. Gallagher and I. Tristani improved this fixed point approach by considering the equation satisfies by f ε −f 0 , where f 0 is known to exist thanks to Theorem 2. This allowed to consider large initial data f in for the Boltzmann equation, and thus large (ρ 0 in , u 0 in , θ 0 in ) for the Navier-Stokes-Fourier system, and in particular, showed that the solution f ε to (1.14) does not blow up before f 0 does.…”

“…-One equation is posed in a "small Gaussian space" and its solution, generated by the macroscopic part Πf in , is shown to converge strongly to a solution of the incompressible Navier-Stokes-Fourier system, by adapting the approach from [5,17]. -The other equation is posed in a "large polynomial space" and describes the evolution of an initial layer generated by the microscopic part f in,⊥ .…”