2014/03/06 by Claudia Garetto, Michael Ruzhansky
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Cauchy problem #Initial value problem #Lie group #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Order (exchange) #Pure mathematics #Sobolev space #Stability and Controllability of Differential Equations #Wave equation #math.AP #math.FA #msc:22E30 #msc:35G10 #msc:35L30 #msc:46F05
paper · pdf · doi:10.1016/j.jde.2015.01.034
published as J. Differential Equations, 258 (2015), 4324-4347 · 21 pages
arxiv created 2014/03/06 · openalex publication_date 2015/02/07 · arxiv updated 2015/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev spaces depending on the order to which the Hörmander condition is satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which the Hörmander condition is satisfied.