Ghost Effect from Boltzmann Theory

Abstract

Taking place naturally in a gas subject to a given wall temperature distribution [Maxwell1879], the ``ghost effect'' exhibits a rare kinetic effect beyond the prediction of classical fluid theory and Fourier law in such a classical problem in physics. As the Knudsen number $\varepsilon$ goes to zero, the finite variation of temperature in the bulk is determined by an $\varepsilon$ infinitesimal, ghost-like velocity field, created by a given finite variation of the tangential wall temperature as predicted by Maxwell's slip boundary condition. Mathematically, such a finite variation leads to the presence of a severe $\varepsilon^{-1}$ singularity and a Knudsen layer approximation in the fundamental energy estimate. Neither difficulty is within the reach of any existing PDE theory on the steady Boltzmann equation in a general 3D bounded domain. Consequently, in spite of the discovery of such a ghost effect from temperature variation in as early as 1960's, its mathematical validity has been a challenging and intriguing open question, causing confusion and suspicion. We settle this open question in affirmative if the temperature variation is small but finite, by developing a new $L^2-L^6-L^{\infty}$ framework with four major innovations: 1) a key $\mathscr{A}$-Hodge decomposition and its corresponding local $\mathscr{A}$-conservation law eliminate the severe $\varepsilon^{-1}$ bulk singularity, leading to a reduced energy estimate; 2) A surprising $\varepsilon^{\frac{1}{2}}$ gain in $L^2$ via momentum conservation and a dual Stokes solution; 3) the $\mathscr{A}$-conservation, energy conservation and a coupled dual Stokes-Poisson solution reduces to an $\varepsilon^{-\frac{1}{2}}$ boundary singularity; 4) a crucial construction of $\varepsilon$-cutoff boundary layer eliminates such boundary singularity via new Hardy and BV estimates.