A priori bounds for the φ⁴ equation in the full sub-critical regime
File(s)
Published version
Author(s)
Weber, Hendrik
Moinat, Augustin
Chandra, Ajay
Type
Journal Article
Abstract
We derive a priori bounds for the Φ4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by
(∂t − )φ = −φ3 + ∞φ + ξ ,
where the term +∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d<4 by adjusting the regularity of the noise term ξ, choosing ξ∈C−3+δ. Our main result states that if ϕ
satisfies this equation on a space–time cylinder D=(0,1)×{|x|⩽1}
, then away from the boundary ∂D
the solution ϕ
can be bounded in terms of a finite number of explicit polynomial expressions in ξ
. The bound holds uniformly over all possible choices of boundary data for ϕ
and thus relies crucially on the super-linear damping effect of the non-linear term −ϕ3
. A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model (Πx)x
and the family of translation operators (Γx,y)x,y
we work with just a single object (Xx,y)
which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.
(∂t − )φ = −φ3 + ∞φ + ξ ,
where the term +∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d<4 by adjusting the regularity of the noise term ξ, choosing ξ∈C−3+δ. Our main result states that if ϕ
satisfies this equation on a space–time cylinder D=(0,1)×{|x|⩽1}
, then away from the boundary ∂D
the solution ϕ
can be bounded in terms of a finite number of explicit polynomial expressions in ξ
. The bound holds uniformly over all possible choices of boundary data for ϕ
and thus relies crucially on the super-linear damping effect of the non-linear term −ϕ3
. A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model (Πx)x
and the family of translation operators (Γx,y)x,y
we work with just a single object (Xx,y)
which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”.
Date Issued
2023-06
Date Acceptance
2023-03-21
Citation
Archive for Rational Mechanics and Analysis, 2023, 247 (3), pp.1-76
ISSN
0003-9527
Publisher
Springer
Start Page
1
End Page
76
Journal / Book Title
Archive for Rational Mechanics and Analysis
Volume
247
Issue
3
Copyright Statement
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Identifier
https://link.springer.com/article/10.1007/s00205-023-01876-7
Publication Status
Published
Article Number
48
Date Publish Online
2023-05-03