2022/06/01 by Chanwoo Kim, Kim, Chanwoo, Joonhyun La +1
Engineering · Mathematics · #35Q20 #35Q31 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Lattice Boltzmann Simulation Studies #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2206.00543
openalex publication_date 2022/06/01 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations St ∂t F + v⋅ ∇x F = (1)/(Kn) Q(F ,F ) toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded ∂t u + u ⋅ ∇x u + ∇x p = 0,div u =0. We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of the hydrodynamic limit toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded (ω∈ L^\mathfrakp for any fixed 1 ≤ \mathfrakp < ∞). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in L^\mathfrakp as \mathfrakp → ∞ (localized Yudovich class).