2023/01/23 by R. Esposito, Yan Guo, Esposito, Raffaele +5
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Lattice Boltzmann Simulation Studies
paper · pdf · doi:10.48550/arxiv.2301.09427
openalex publication_date 2023/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ε goes to zero, the finite variation of temperature in the bulk is determined by an ε 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 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 L2-L6-L∞ framework with four major innovations: 1) a key \mathscrA-Hodge decomposition and its corresponding local \mathscrA-conservation law eliminate the severe ε-1 bulk singularity, leading to a reduced energy estimate; 2) A surprising ε(1)/(2) gain in L2 via momentum conservation and a dual Stokes solution; 3) the \mathscrA-conservation, energy conservation and a coupled dual Stokes-Poisson solution reduces to an ε-(1)/(2) boundary singularity; 4) a crucial construction of ε-cutoff boundary layer eliminates such boundary singularity via new Hardy and BV estimates.