2021/07/09 by Francesca Anceschi, Anceschi, Francesca, Annalaura Rebucci +1
Mathematics · #35B09 #35B45 (secondary) #35B65 #35H20 #35K70 (Primary) 35K65 #35Q84 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2107.04441
openalex publication_date 2021/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation \mathscrL u = f with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space W, suitable for the study of weak solutions to \mathscrLu = f, that allows us to prove a weak Poincaré inequality. More precisely, our goal is to prove a weak Harnack inequality for non-negative super-solutions by considering their Log-transform and following S. N. Kruzkov (1963). Then this functional inequality is combined with a classical covering argument (Ink-Spots Theorem) that we extend for the fist time to the case of ultraparabolic equations.