On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces

2023/11/08 by Pascal Auscher, Auscher, Pascal, Hedong Hou +1
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary 35K45 #Secondary 42B37

paper · pdf · doi:10.48550/arxiv.2311.04844

openalex publication_date 2023/11/08 · openalex created_date 2023/11/10 · openalex updated_date 2026/07/28

Abstract

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem ∂_t u - div A ∇ u = f, u(0)=0, where the source f also lies in (different) weighted tent spaces, provided the complex coefficient matrix A is bounded, measurable, time-independent, and uniformly elliptic. To achieve this, we extend the theory of singular integral operators on tent spaces via off-diagonal estimates introduced by [arXiv:1112.4292] to obtain estimates on solutions u, and also ∇ u, ∂_t u, and div A ∇ u in weighted tent spaces, showing at the same time maximal regularity. Uniqueness follows from a different strategy using interior representation for weak solutions and boundary behavior.

Citations

• Some new function spaces and their applications to Harmonic analysis

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