Blanc SIMI 1 - Blanc - SIMI 1 - Mathématiques et interactions

Asymptotic analysis of Evolution Partial Differential Equations – ANAÉ

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 . ( 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.

Submission summary

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.


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

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