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

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

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. We remark that this solution differs from a local equilibrium Maxwellian, hence it is a genuine non-equilibrium stationary solution. 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. This talk is based on the joint work with Esposito, Guo, Marra.