TY - JOUR
T1 - Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma
AU - Gerasimovičs, Andris
AU - Hocquet, Antoine
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.
KW - Multiplicative Sewing lemma
KW - Propagator
KW - Rough partial differential equations
KW - Rough path
UR - http://www.scopus.com/inward/record.url?scp=85112282312&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2021.109200
DO - 10.1016/j.jfa.2021.109200
M3 - Article
AN - SCOPUS:85112282312
SN - 0022-1236
VL - 281
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 10
M1 - 109200
ER -