Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

Andris Gerasimovičs; Antoine Hocquet; Torstein Nilssen

doi:10.1016/j.jfa.2021.109200

Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

Andris Gerasimovičs, Antoine Hocquet, Torstein Nilssen

Department of Mathematical Sciences

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Abstract

We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form du_t−L_tu_tdt=N(u_t)dt+∑_i=1^dF_i(u_t)dX_tⁱ where (L_t)_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 language	English
Article number	109200
Journal	Journal of Functional Analysis
Volume	281
Issue number	10
Early online date	28 Jul 2021
DOIs	https://doi.org/10.1016/j.jfa.2021.109200
Publication status	Published - 15 Nov 2021

Bibliographical note

Funding Information:
The authors would like to thank M. Hairer for many useful discussions and comments. The anonymous referees are warmly thanked for their careful reading of the paper, notably for having pointed out several gaps in the first version. AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize (grant number RL-2012-020 ). AH and TN gratefully acknowledge the financial support by the DFG via Research Unit FOR 2402 (277012070).

Keywords

Multiplicative Sewing lemma
Propagator
Rough partial differential equations
Rough path

ASJC Scopus subject areas

Analysis

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multiplicative_JFA_revision7_as_submitted.PDFAccepted author manuscript, 746 KBLicence: CC BY-NC-ND

Cite this

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title = "Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma",

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 H{\"o}lder 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.",

keywords = "Multiplicative Sewing lemma, Propagator, Rough partial differential equations, Rough path",

author = "Andris Gerasimovi{\v c}s and Antoine Hocquet and Torstein Nilssen",

note = "Funding Information: The authors would like to thank M. Hairer for many useful discussions and comments. The anonymous referees are warmly thanked for their careful reading of the paper, notably for having pointed out several gaps in the first version. AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize (grant number RL-2012-020 ). AH and TN gratefully acknowledge the financial support by the DFG via Research Unit FOR 2402 (277012070). ",

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AU - Nilssen, Torstein

N1 - Funding Information: The authors would like to thank M. Hairer for many useful discussions and comments. The anonymous referees are warmly thanked for their careful reading of the paper, notably for having pointed out several gaps in the first version. AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize (grant number RL-2012-020 ). AH and TN gratefully acknowledge the financial support by the DFG via Research Unit FOR 2402 (277012070).

PY - 2021/11/15

Y1 - 2021/11/15

N2 - 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.

AB - 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.

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Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

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Bibliographical note

Keywords

ASJC Scopus subject areas

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