Ghost effect from Boltzmann theory

Taking place naturally in a gas subject to a given wall temperature distribution, 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 goes to zero, the finite variation of temperature in the bulk is determined by an 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 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 1960s, 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 framework with four major innovations as follows: (1) a key ‐Hodge decomposition and its corresponding local ‐conservation law eliminate the severe bulk singularity, leading to a reduced energy estimate; (2) a surprising gain in via momentum conservation and a dual Stokes solution; (3) the ‐conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an boundary singularity; (4) a crucial construction of ‐cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates.