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Ghost effect from Boltzmann theory
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DOI：10.1002/cpa.70017.png)
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Abstract
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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 ε $\varepsilon$ goes to zero, the finite variation of temperature in the bulk is determined by an ε $\varepsilon$ 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 ε − 1 $\varepsilon ^{-1}$ 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 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 L 2 − L 6 − L ∞ $L^2-L^6-L^\infty$ framework with four major innovations as follows: (1) a key A $\mathcal {A}$ -Hodge decomposition and its corresponding local A $\mathcal {A}$ -conservation law eliminate the severe ε − 1 $\varepsilon ^{-1}$ bulk singularity, leading to a reduced energy estimate; (2) a surprising ε 1 2 $\varepsilon ^{\frac{1}{2}}$ gain in L 2 $L^2$ via momentum conservation and a dual Stokes solution; (3) the A $\mathcal {A}$ -conservation, energy conservation, and a coupled dual Stokes–Poisson solution reduces to an ε − 1 2 $\varepsilon ^{-\frac{1}{2}}$ boundary singularity; (4) a crucial construction of ε $\varepsilon$ -cutoff boundary layer eliminates such boundary singularity via new Hardy's and BV estimates.
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