Space regularity for evolution operators modeled on H "ormander vector\n fields with time dependent measurable coefficients

2019/03/18 by Marco Bramanti, Bramanti, Marco
Mathematics · #35R03 (Primary) 35B65 #35R05 (Secondary) #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1903.07327

openalex publication_date 2019/03/18 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

We consider a heat-type operator L structured on the left invariant\n1-homogeneous vector fields which are generators of a Carnot group, multiplied\nby a uniformly positive matrix of bounded measurable coefficients depending\nonly on time. We prove that if Lu is smooth with respect to the space\nvariables, the same is true for u, with quantitative regularity estimates in\nthe scale of Sobolev spaces defined by right invariant vector fields. Moreover,\nthe solution and its space derivatives satisfy a 1/2-H "older continuity\nestimate with respect to time. The result is proved both for weak solutions and\nfor distributional solutions, in a suitable sense.\n

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