Graduate Seminar (S4B1)
Renormalization Group and nonlinear PDEs
Prof. Dr. M. Disertori, Dr. M. Lohmann
Organization.
Seminar sessions Tuesdays 8.30–10, room 2.040
In case of question please contact
M. Lohmann lohmann(at)iam.uni-bonn.de (office 4.036)
M. Disertori disertori(at)iam.uni-bonn.de (office 4.045)
Tentative schedule.
- (19.04) Introduction M. Lohmann
- (26.04) Non-linear heat equation (Part 1) L. Lechner
- (03.05) Non-linear heat equation (Part 2) L. Borasi
- (10.05) Stochastic versus deterministic, infrared versus ultraviolet N. Barashkov
- (31.05) Linearized renormalization group T. Friesel
- (07.06) Second order perturbation theory S. Hilger or M. Lager
- (14.06) Renormalized powers for white noise J. Jansen
- (21.06) Fixed point problem U. Adil
- (31.05) Completing the proof M. Lager or S. Hilger
Synopsis.
We consider nonlinear parabolic PDE’s of the form
E : ∂t u = ∆u + F(u),
where u=u(t,x), t≥0, x∈ Λ ⊂ Rd , the nonlinear function F(u) depends on u (and possibly some derivatives), and may contain a random part. These equations are ubiquitous on physics. Two examples are the effect of temperature on fluid flows and random deposition in surface growth.
In physics one is often interested in the long time/space behavior of solutions. For that one introduces a mollified version Eε of the equation, where is a small scale regularization. The large scale behavior should not depend on the choice of the mollifier. From a mathematical point of view (and sometimes also in physics), one wants to understand well-posedness of E i.e., the small time/space behavior of the solution u. The problem is that u is expected to have very weak regularity (it is a distribution) and it is not clear how to set up the solution theory in distribution space. The strategy is to consider the regularized equation Eε and to study the limit ε→ 0. Recently this problem was addressed by Hairer who set up a solution theory for a class of such equations.
In this seminar we plan to consider an alternative approach. This is the so-called rigorous renormalization group analysis, inspired by techniques in theoretical physics. The idea is to reformulate the problem as a fixed point problem in a Banach space. We consider a growing sequence of scales εn= ε 2n and construct a sequence of effective equations Eεn . We stop as soon as εn =1. The strategy is then to tune the parameters in the starting equation E=E0 in order for Eεn to have a nice limit as ε→ 0 (equivalent to n → ∞).
Literature.
To understand the RG strategy we will start by reading part of the paper (dealing with the problem of long time/space behavior)
-'Renormalization group and asymptotics of solutions of nonlinear parabolic equations' J. Bricmont, A. Kupiainen, G. Lin
Comm. Pure, Appl. Math., 47 (1994), pp. 893–921
We will then proceed with
-'Renormalization Group and Stochastic PDEs', Antti Kupiainen
Ann. Henri Poincare 17 (2016), 497–535