Institute for applied mathematics

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.

 

  1. (19.04)   Introduction  M. Lohmann
  2. (26.04)   Non-linear heat equation  (Part 1)   L. Lechner
  3. (03.05)   Non-linear heat equation  (Part 2)   L. Borasi
  4. (10.05)   Stochastic versus deterministic, infrared versus ultraviolet                                           N.  Barashkov
  5. (31.05)   Linearized renormalization group     T.  Friesel
  6. (07.06)   Second order perturbation theory     S.  Hilger or M. Lager
  7. (14.06)   Renormalized powers for white noise      J.  Jansen
  8. (21.06)   Fixed point problem     U.  Adil
  9. (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  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

 

 

 

 

News

Prof. Dr. S. Müller erhält den diesjährigen Lehrpreis der Universität Bonn (07.07.17).

Prof. Dr. Michael Ortiz kommt als Bonn Research Chair ans IAM (Pressemitteilung, 29.7.2016).

Prof. Dr. S. Conti erhält den diesjährigen Lehrpreis der Universität Bonn (05.07.2016).

Prof. Dr. Karl-Theodor Sturm, Koordinator des Hausdorff Zentrums für Mathematik an der Universität Bonn, erhält für seine eigene Forschung einen begehrten Advanced Grant des Europäischen Forschungsrats (ERC). (Pressemitteilung 21.04.2016)

Contact

Managing Director: Prof. Dr. Anton Bovier
Chief Administrator: Dr. B. Doerffel
geschaeftsfuehrung@iam.uni-bonn.de
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Mailing address

Institute for Applied Mathematics
University of Bonn
Endenicher Allee 60
D-53115 Bonn / Germany