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Given an obstacle in ℝ3 and a non-zero velocity with small amplitude at the infinity, we construct the unique steady Boltzmann solution flowing around such an obstacle with the prescribed velocity as |𝑥|→∞ , which approaches the corresponding Navier–Stokes steady flow, as the mean-free path goes to zero. Furthermore, we establish the error estimate between the Boltzmann solution and its Navier–Stokes approximation. Our method consists of new L6 and L3 estimates in the unbounded exterior domain, as well as an iterative scheme preserving the positivity of the distribution function.
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Stationary solutions to the Boltzmann equation in the hydrodynamic limit. Ann. PDE 4 (2018), no. 1, Art. 1, 119 pp.
Abstract Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition.We employ a recent quantitative L2–L∞ approach with new L6 estimates for the hydrodynamic part P f of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotic stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system.
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- Award ID(s):
- 1810868
- PAR ID:
- 10093619
- Date Published:
- Journal Name:
- Annals of PDE
- Volume:
- 4
- Issue:
- no.1
- Page Range / eLocation ID:
- Art, 119pp
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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