TY - JOUR

T1 - The Dynamic Φ34 Model Comes Down from Infinity

AU - Mourrat, Jean Christophe

AU - Weber, Hendrik

N1 - Funding Information:
Acknowledgement. JCM is partially supported by the ANR Grant LSD (ANR-15-CE40-0020-03). HW acknowledges support by an EPSRC First Grant, a Royal Society University Research Fellowship and the Mathematical Sciences Research Institute where part of this work was completed.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - We prove an a priori bound for the dynamic Φ34 model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov–Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean Φ34 field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

AB - We prove an a priori bound for the dynamic Φ34 model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows one to construct invariant measures via the Krylov–Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean Φ34 field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

UR - http://www.scopus.com/inward/record.url?scp=85030835247&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030835247&partnerID=8YFLogxK

U2 - 10.1007/s00220-017-2997-4

DO - 10.1007/s00220-017-2997-4

M3 - Article

AN - SCOPUS:85030835247

VL - 356

SP - 673

EP - 753

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -