Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

Andris Gerasimovičs, Antoine Hocquet, Torstein Nilssen

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form dut−Ltutdt=N(ut)dt+∑i=1dFi(ut)dXti where (Lt)t∈[0,T] is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while X≡(X,X) is a two-step (non-necessarily geometric) rough path of Hölder regularity γ>1/3. Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path X). As a technical tool, we introduce a version of the multiplicative sewing lemma, which allows to construct the so called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.

Original languageEnglish
Article number109200
JournalJournal of Functional Analysis
Volume281
Issue number10
Early online date28 Jul 2021
DOIs
Publication statusE-pub ahead of print - 28 Jul 2021

Keywords

  • Multiplicative Sewing lemma
  • Propagator
  • Rough partial differential equations
  • Rough path

ASJC Scopus subject areas

  • Analysis

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