2020/05/25 by Juhi Jang, Chanwoo Kim, Jang, Juhi +1 · 4 citations
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.2005.12192
77 pages, submitted
arxiv created 2020/05/25 · openalex publication_date 2020/05/25 · arxiv updated 2020/05/26 · openalex created_date 2020/05/29 · openalex updated_date 2026/07/28
A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance we capture the Navier-Stokes equations of viscosity ∼ (Knudsen number)/(Mach number) satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative Lp-L^∞ estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green's function approach. The incompressible Euler limit follows as a byproduct of our framework.