Communications in Mathematical Physics

, Volume 323, Issue 1, pp 177–239 | Cite as

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law



In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem:
$$v \cdot \nabla _{x}F =\frac{1}{{\rm K}_{\rm n}}Q(F,F),\qquad (x,v)\in \Omega \times \mathbf{R}^{3}, \quad \quad (0.1) $$
$$F(x,v)|_{n(x)\cdot v<0} = \mu _{\theta}\int_{n(x) \cdot v^{\prime}>0}F(x,v^{\prime})(n(x)\cdot v^{\prime})dv^{\prime},\quad x \in\partial \Omega,\quad \quad (0.2) $$
where Ω is a bounded domain in \({\mathbf{R}^{d}, 1 \leq d \leq 3}\), Kn is the Knudsen number and \({\mu _{\theta}=\frac{1}{2\pi \theta ^{2}(x)} {\rm exp} [-\frac{|v|^{2}}{2\theta (x)}]}\) is a Maxwellian with non-constant(non-isothermal) wall temperature θ(x). Based on new constructive coercivity estimates for both steady and dynamic cases, for \({|\theta -\theta_{0}|\leq \delta \ll 1}\) and any fixed value of Kn, we construct a unique non-negative solution F s to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non-equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion \({F_s=\mu_{\theta_0}+\delta F_{1}+O(\delta ^{2})}\) and we prove that, if the Fourier law holds, the temperature contribution associated to F 1 must be linear, in the slab geometry.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.International Research Center M&MOCSUniv. dell’AquilaCisterna di LatinaItaly
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA
  3. 3.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  4. 4.Dipartimento di Fisica and Unità INFNUniversità di Roma Tor VergataRomaItaly

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