Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

Monday, November 5, 2012 -
3:15pm to 4:15pm
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.
Chanwoo Kim
Event Location: 
Fine Hall 314