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

Raffaele Esposito, Yan Guo, Chanwoo Kim, Rossana Marra
In the study of the heat transfer in the Boltzmann theory, the basic problem
is to construct solutions to the steady problem for the Boltzmann equation in a
general bounded domain with diffuse reflection boundary conditions
corresponding to a non isothermal temperature of the wall. Denoted by \delta
the size of the temperature oscillations on the boundary, we develop a theory
to characterize such a solution mathematically. We construct a unique solution
F_s to the Boltzmann equation, which is dynamically asymptotically stable with
exponential decay rate. Moreover, if the domain is convex and the temperature
of the wall is continuous we show that F_s is continuous away from the grazing
set. If the domain is non-convex, discontinuities can form and then propagate
along the forward characteristics. We show that they actually form for a
suitable smooth temperature profile. We remark that this solution differs from
a local equilibrium Maxwellian, hence it is a genuine non equilibrium
stationary solution. Our analysis is based on recent studies of the boundary
value problems for the Boltzmann equation but with new constructive coercivity
estimates for both steady and dynamic cases. A natural question in this setup
is to determine if the general Fourier law, stating that the heat conduction
vector q is proportional to the temperature gradient, is valid. As an
application of our result we establish an expansion in \delta for F_s whose
first 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 of
F_1 must be linear in a slab. Such a necessary condition contradicts available
numerical simulations, leading to the prediction of break-down of the Fourier
law in the kinetic regime.
View original: http://arxiv.org/abs/1111.5843