Asymptotic analysis of Evolution Partial Differential Equations – ANAÉ
Asymptotic Analysis of Evolution Equations
<br />The goal of this project is to blend together ideas from these three points of view, to gain some new insights on the behaviour of solutions of partial differential equations in some asymptotic regimes : <br /><br /> Long time behaviour, <br />Existence, scattering,<br />Stabilities, instabilities,<br />Blow-up<br />
PDE's DYNAMICS AND STATISTICAL APPROACHES
During the last decades, the interface between two very active fields of research in mathematics, non-linear<br />partial differential equations and infinite dimensional dynamical systems has been growing rapidly. Indeed, the numerous advances in non linear partial differential equations allow some perspectives which appeared to be completely out of reach a few years ago and open the very exciting perspective of studying deep dynamical properties of solutions of Partial differential equations. The project aims at gathering a group of French<br />researchers, from diverse backgrounds, already involved in the study of non linear partial differential equations and dynamical systems, and boosting their activity on those matters<br />through interactions, not only among the participants to this ANR, but also with the community at<br />large and particularly at the international level. Major advances have been made on the study in the large of solutions of non-linear evolution <br />partial differential equations exploiting Hamiltonian structures. One can mention the construction of periodic or<br />quasi-periodic solutions, the study of solutions to the Cauchy problem on long time intervals, the discovery of new<br />integrable PDEs, the use of randomness to reveal surprising new phenomena. Most of the advances have shown how fruitful <br />the cross-fertilization of ideas originating from dynamical systems, non-linear PDEs (Birkhoff normal forms methods,<br />microlocal and semi-classical analysis,\dots) and/or a statistical point of view can be. The originality of the<br />present project is to blend together persons having these diverse expertises together with recognized specialists of partial<br />differential equations.<br />
As presented in the title of the proposal, the guidelines of our project are the study of {\em partial differential equations} in some {asymptotic regimes}. The aims are to obtain on various topics deep descriptions of the dynamical behaviour of solutions (which means that in some cases we will have in a first step to prove the existence of these solutions). The originality of our project is that we intend to do so by blending together ideas coming from different fields such as dynamical systems (through KAM, small divisors and normal forms theories for example) or probabilities, by studying some statistical quantities rather than using the more precise (but less amenable) deterministic approach.
Since the beginning of the contract, Jean-Marc Delort worked on the Klein-Gordon equation semilinear modified small and regular data.
Hatem Zaag continued several works relating to explosion problems for waves or heat equations .
For Nantes partner:
Thomann worked on the modified scattering for Schrodinger equation
He also developed with Burq Tzvetkov a new approach to build low probabilistic solutions, using a Gibbs measure.
Grébert KAM has developed the KAM method for dispersive equations
For the Paris-Sud partner
In collaboration with Sandrine Grellier,
P. Gerard continued the study of the cubic Szego equation.
They got two types of results:
- Construction of an inverse spectral transformation, taking into account
Spectral multiplicity phenomena,
- Application to the low turbulence: Building solutions with unbounded large indexes Sobolev norms.
H. Xu completed her thesis
On some Hamiltonian systems related to the cubic equation Szego, «
under the direction of P. GERARD
Alazard T., N. Burq C.Zuily developed the semi-classical approach for obtaining dispersion equation estimations for the water-waves
N. Burq C.Zuily were also interested in understanding the effects of concentrations eigenfunctions of the Laplace operator on tori (or more generally quasi-modes) to obtain new results of stabilization
N. Burq (with R. Joly) also obtained new results for the stabilization waves in unbounded domain.
N.Burq (with G. and W. Schlag Raugel) were interested in the dynamics in time to the Klein-Gordon equation unamortized linear.
Alazard got new control results for the equation of water-waves.
1) Szego cubic equation and low turbulence : construction of solutions with large indexes Sobolev norms unbounded. Research news from INSMI . www.cnrs.fr/insmi/spip.php ( Gerard , Grellier )
2 ) water - waves Control ( Alazard , Baldi , Han- Kwan )
3) Large time dynamic for the damped nonlinear Klein -Gordon equation ( Burq , Raugel , Schlag )
1. arXiv:1507.02307 N. Burq, D.Dos Santos, K. Krupchyk
2. arXiv:1505.05981 N. Burq , G. Raugel , W. Schlag
3. arXiv:1503.05513 N Burq , C. Zuily
4. arXiv:1503.02058 N. Burq , C. Zuily
5. arXiv:1412.7499N. Burq , L. Thomann Nikolay Tzvetkov
6. arXiv:1408.6054 N. Burq, Romain Joly
7. arXiv:1506.08520 T. Alazard
8. arXiv:1501.06366 T. Alazard, P. Baldi, Daniel Han-Kwan
9. arXiv:1405.1934T. Alazard, P. Baldi
10. arXiv:1509.03520 V.T. Nguyen, H. Zaag
11. arXiv:1506.08306 S. Tayachi, H. Zaag
12. arXiv:1506.07708 FM., N. Nouaili, H. Zaag
13. arXiv:1410.4079 VT. Nguyen, H. Zaag
14. arXiv:1406.5233 VT., H. Zaag
15. arXiv:1509.06873 T. Oh, G. Richards, L. Thomann (IECL)
16. arXiv:1509.02093 T. Oh, L. Thomann (IECL)
17. arXiv:1505.01698 F. Hérau (LMJL), L. Thomann (LMJL)
18. arXiv:1502.07699 B. Grébert (LMJL), E. Paturel (LMJL), L. Thomann (LMJL)
19. arXiv:1502.05643 Pierre Germain (CIMS), Zaher Hani (GATECH), L. Thomann (LMJL)
20. arXiv:1501.03760 P. Germain (CIMS), Z. Hani (GATECH), L. Thomann (LMJL)
21. arXiv:1408.6213 . Hani (GATECH), L. Thomann (LMJL)
22. arXiv:1403.4913 R. Imekraz (IMB), Didier Robert (LMJL), L. Thomann (LMJL)
23. arXiv:1502.07699 B. Grébert (LMJL), E. Paturel (LMJL), L. Thomann (LMJL)
24. arXiv:1508.06814 S. Grellier , Patrick Gerard
25. arXiv:1402.1716 . Gerard , S. Grellier
26. hal-00945805v1 JM. Delort.
The last decades has witnessed a very fast and deep development in the field of evolution partial differential equations (and particularly the dispersive equations). These major advances allow some perspectives which appeared to be completely out of reach a few years ago and open the very exciting perspective of studying deep dynamical properties of solutions of Partial differential equations. On the other hand, specialists of dynamical systems successfully extended methods and ideas, developed for the study of finite dimensional models, to infinite dimensional ones. This led to spectacular results concerning long time behavior of solutions of some non-linear partial differential equations, especially in one space dimension. The tools developed in the PDE context to handle non-linear PDEs could also lead to major breakthroughs, when combined with some dynamical properties of the equations. Finally, the study of partial differential equations in the presence of randomness, a topic originating in ideas from statistical mechanics, has also recently seen spectacular results.
The goal of this project is to blend together ideas from these three points of view, to gain some new insights on the behaviour of solutions of partial differential equations in some asymptotic regimes :
-- Long time behaviour,
-- Existence, scattering,
-- Stabilities, instabilities of particular solutions
-- Blow-up.
Project coordination
Nicolas BURQ (Laboratoire de mathématiques d'Orsay)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
Partnership
LMO UMR 8628 Laboratoire de mathématiques d'Orsay
LAGA UMR 7539 LAGA Université Paris-Nord
LMJL UMR 6629 Laboratoire de mathématiques Jean Leray, Université de Nantes
Help of the ANR 245,000 euros
Beginning and duration of the scientific project:
December 2013
- 48 Months
