On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

2024/06/22 by Auscher, Pascal, Hou, Hedong
#35K05 #42B30 #46E35 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 35K45 #Secondary 42B37

paper · doi:10.48550/arxiv.2406.15775

Abstract

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of p for which tempered distributions in homogeneous Hardy--Sobolev spaces H^s,p with regularity index s ∈ (-1,1) are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces B^s_p,p.

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