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In the study of the heat transfer in the Boltzmann theory, the basic problemis to construct solutions to the steady problem for the Boltzmann equation in ageneral bounded domain with diffuse reflection boundary conditionscorresponding to a non isothermal temperature of the wall. Denoted by \deltathe size of the temperature oscillations on the boundary, we develop a theoryto characterize such a solution mathematically. We construct a unique solutionF_s to the Boltzmann equation, which is dynamically asymptotically stable withexponential decay rate. Moreover, if the domain is convex and the temperatureof the wall is continuous we show that F_s is continuous away from the grazingset. If the domain is non-convex, discontinuities can form and then propagatealong the forward characteristics. We show that they actually form for asuitable smooth temperature profile. We remark that this solution differs froma local equilibrium Maxwellian, hence it is a genuine non equilibriumstationary solution. Our analysis is based on recent studies of the boundaryvalue problems for the Boltzmann equation but with new constructive coercivityestimates for both steady and dynamic cases. A natural question in this setupis to determine if the general Fourier law, stating that the heat conductionvector q is proportional to the temperature gradient, is valid. As anapplication of our result we establish an expansion in \delta for F_s whosefirst order term F_1 satisfies a linear, parameter free equation. Consequently,we discover that if the Fourier law were valid for F_s, then the temperature ofF_1 must be linear in a slab. Such a necessary condition contradicts availablenumerical simulations, leading to the prediction of break-down of the Fourierlaw in the kinetic regime.

Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory