2010/05/10 by Diogo Arsénio, Arsénio, Diogo, Laure Saint‐Raymond +2
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Numerical methods in inverse problems #math.AP #math.FA
paper · pdf · doi:10.48550/arxiv.1005.1547
31 pages, 1 figure Paper withdrawn. This is an old obsolete version of the article
openalex publication_date 2010/05/10 · arxiv created 2012/06/28 · arxiv updated 2012/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family fλ(x,v)∈ Lp satisfying some appropriate transport relation v⋅∇x fλ= (1-Δx)^\fracβ2(1-Δv)^\fracα2gλ may be inferred solely from its compactness in v. This method is introduced as an alternative to the lack of known suitable averaging lemmas in L1 when the right-hand side of the transport equation has very low regularity, due to an external force field for instance. In a forthcoming work, the authors make a crucial application of this new approach to the study of the hydrodynamic limit of the Boltzmann equation with a rough force field.