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: $$v \cdot \nabla _{x}F =\frac{1}{{\rm K}_{\rm n}}Q(F,F),\qquad (x,v)\in \Omega \times \mathbf{R}^{3}, \quad \quad (0.1) $$ v · ? x F = 1 K n Q ( F , F ) , ( x , v ) ∈ Ω × R 3 , ( 0.1 ) $$F(x,v)|_{n(x)\cdot v<0} = \mu _{\theta}\int_{n(x) \cdot v^{\prime}>0}F(x,v^{\prime})(n(x)\cdot v^{\prime})dv^{\prime},\quad x \in\partial \Omega,\quad \quad (0.2) $$ F ( x , v ) | n ( x ) · v < 0 = μ θ ∫ n ( x ) · v ′ > 0 F ( x , v ′ ) ( n ( x ) · v ′ ) d v ′ , x ∈ ? Ω , ( 0.2 ) where Ω is a bounded domain in ${\mathbf{R}^{d}, 1 \leq d \leq 3}$ R d , 1 ≤ d ≤ 3 , K_n is the Knudsen number and ${\mu _{\theta}=\frac{1}{2\pi \theta ^{2}(x)} {\rm exp} -\frac{|v|^{2}}{2\theta (x)}]}$ μ θ = 1 2 π θ 2 ( x ) exp - | v | 2 2 θ ( 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 ${|\theta -\theta_{0}|\leq \delta \ll 1}$ | θ - θ 0 | ≤ δ ? 1 and any fixed value of K_n, 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 ${F_s=\mu_{\theta_0}+\delta F_{1}+O(\delta ^{2})}$ F s = μ θ 0 + δ F 1 + O ( δ 2 ) and we prove that, if the Fourier law holds, the temperature contribution associated to F ₁ must be linear, in the slab geometry.