Communications in Mathematical Physics

, Volume 323, Issue 1, pp 177–239

# Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

Article

## Abstract

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 Fs 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 F1 must be linear, in the slab geometry.