Boltzmann diffusive limit beyond the Navier-Stokes approximation. (English) Zbl 1089.76052
Summary: Given a normalized Maxwellian \(\mu\) and \(n\geq 1\), we establish the global-in-time validity of a diffusive expansion
\[
F^\varepsilon(t,x,v)= \mu+ \sqrt{\mu} \{\varepsilon f_1(t,x,v)+ \varepsilon^2 f_2(t,x,v)+\cdots+ \varepsilon^n f_n^\varepsilon (t,x,v)\},
\]
for a solution \(F^\varepsilon\) to the rescaled Boltzmann equation (diffusive scaling)
\[
\varepsilon\partial_t F^\varepsilon+ v\cdot\nabla_x F^\varepsilon= \tfrac 1\varepsilon Q(F^\varepsilon, F^\varepsilon)
\]
inside a periodic box \(\mathbb T^3\). We assume that in the initial diffusive expansion at \(t=0\), the fluid parts of these \(f_m(0,x,v)\) have arbitrary divergence-free velocity fields as well as temperature fields for all \(1\leq m\leq n\) while \(f_1(0,x,v)\) has small amplitude in \(H^2\). For \(m\geq 2\), these \(f_m(t,x,v)\) are determined by a sequence of linear Navier-Stokes-Fourier systems iteratively. More importantly, the remainder \(f_n^\varepsilon(t,x,v)\) is proven to decay in time uniformly in \(\varepsilon\) via a unified nonlinear energy method. In particular, our results lead to an error estimate for \(f_1(t,x,v)\), the well-known Navier-Stokes-Fourier approximation, and beyond. The collision kernel \(Q\) includes hard-sphere, the cutoff inverse-power, as well as the Coulomb interactions.
MSC:
| 76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |
| 76M45 | Asymptotic methods, singular perturbations applied to problems in fluid mechanics |
| 82B40 | Kinetic theory of gases in equilibrium statistical mechanics |
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