Fourier Law and Non-Isothermal Boundary in the Boltzmann Theory

TL;DR

This paper constructs and analyzes solutions to the Boltzmann equation in bounded domains with non-isothermal boundaries, revealing conditions under which Fourier's law breaks down in the kinetic regime.

Contribution

It develops a mathematical theory for steady solutions with non-isothermal boundaries, including stability, continuity, and discontinuity analysis, and challenges the validity of Fourier's law in this context.

Findings

Constructed unique, stable steady solutions to the Boltzmann equation with non-isothermal boundaries.

Showed solutions are continuous in convex domains but can have propagating discontinuities in non-convex domains.

Discovered that Fourier's law does not hold in the kinetic regime based on the behavior of the first-order correction.

Abstract

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.