Analysis of Geometrical Effects on Dispersive Equations

Project Information

ANADEL

Grant agreement ID: 757996

DOI

10.3030/757996

Project closed

EC signature date 22 September 2017

Start date 1 February 2018

End date 31 July 2024

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 293 763,00

EU contribution

€ 1 293 763,00

Coordinated by

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE CNRS
France

Periodic Reporting for period 4 - ANADEL (Analysis of Geometrical Effects on Dispersive Equations)

Reporting period: 2022-08-01 to 2024-07-31

Summary of the context and overall objectives of the project

We study the decay properties of waves are closely related to the structure and dispersion of the Hamilton flow. Because of this, one can distinguish two fundamentally different situations, namely that of concave or convex domains. In the first case, the reflection points are unique, and the problem can be thought of as a long-time problem; a good model is then the exterior of a convex set. In the second case, one may have a very large number of reflections over a short time, and the problem can be primarily thought of as a local problem.
The project was naturally subdivided into two major focus areas, each with its own overarching objective. These major objectives were :
o Objective 1 : Wave and / or Schrödinger equations in exterior domains (i.e. exterior of a convex - a (weakly) non-trapping domain, etc) ; in this case we wanted to understand: sharp dispersive / Strichartz estimates; diffractive effects.
o Objective 2 : Wave and / or Schrödinger equations in bounded domains (inside a convex - or inside a general bounded domain with points on the boundary where the curvature changes its sign) ; in this case we wanted to understand : sharp dispersive / Strichartz estimates (spectral projector bounds; concentration of eignemodes, etc). Included in this topic are issues related to nonlinear long-time dynamics of dispersive equations on compact, boundary-less manifolds, with the aim of developing robust enough tools that can be readily carried to boundary value problems.
• Objective 3 : Exponential sums estimates and relation to parametrix construction for the Schrödinger equation on domains : so far, number theoretic methods have been a key tool for developing PDEs on compact manifolds.

All addressed problems in ANADEL are fundamental and long-standing problems ; essentially all of them required new methods and approaches. Important advances on the first two objectives have been achieved during the duration of the project and partial results have been obtained by the PI on Objective 3. Besides, some of the techniques we developed may yield advances for dispersive equations on domains without boundary.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

[16] Iandoli, Felice(I-CLBR-MI); Ivanovici, Oana(F-SORBU-LJL)
Dispersion for the wave equation outside a cylinder in ℝ3.
J. Funct. Anal.286(2024) no.9 Paper No. 110377, 50 pp.

[15] Iandoli, Felice(I-CLBR-NDM); Niu, Jingrui(F-SORBU-LJL)
Controllability of quasi-linear Hamiltonian Schrödinger equations on tori.
J. Differential Equations390(2024), 125–170.

[14] Ivanovici, Oana(F-SORBU-LJL); Lascar, Richard(F-NICE-LD); Lebeau, Gilles(F-NICE-LD); Planchon, Fabrice(F-SORBU-IMJ)
Dispersion for the wave equation inside strictly convex domains II: The general case.
Ann. PDE9(2023), no.2 Paper No. 14, 117 pp.

[13] Ivanovici, Oana(F-SORBU-LJL)
Dispersive estimates for the Schrödinger equation in a model convex domain and applications.
Ann. Inst. H. Poincaré C Anal. Non Linéaire40(2023), no.4 959–1008.

[12] Planchon, Fabrice(F-SORBU-IMJ); Tzvetkov, Nikolay(F-ENSLY-PM); Visciglia, Nicola(I-PISA)
Growth of Sobolev norms for 2d NLS with harmonic potential.(English summary)
Rev. Mat. Iberoam.39(2023) no.4 1405–1436.

[11] Planchon, Fabrice(F-SORBU-IMJ); Tzvetkov, Nikolay(F-ENSLY-PM); Visciglia, Nicola(I-PISA)
Modified energies for the periodic generalized KdV equation and applications.
Ann. Inst. H. Poincaré C Anal. Non Linéaire40(2023), no.4 863–917.


[10] Feola, Roberto(I-ROME3-MP); Grébert, Benoît(F-NANT-LM); Iandoli, Felice(I-CLBR-MI)
Long time solutions for quasilinear Hamiltonian perturbations of Schrödinger and Klein-Gordon equations on tori.
Anal. PDE16(2023), no.5 1133–1203.


[9] Iandoli, Felice(I-CLBR-MI)
On the Cauchy problem for quasi-linear Hamiltonian KdV-type equations.Qualitative properties of dispersive PDEs, 167–186.
Springer INdAM Ser., 52 Springer, Singapore, [2022], ©2022 ISBN:978-981-19-6433-6 ISBN:978-981-19-6434-3
Part of Book Collection MR4628069

[8] Feola, Roberto(I-MILAN); Iandoli, Felice(F-SORBU-LJL); Murgante, Federico(I-SISSA-NDM)
Long-time stability of the quantum hydrodynamic system on irrational tori.
Math. Eng.4(2022) no.3 Paper No. 023, 24 pp.

[7] Feola, Roberto(I-MILAN); Iandoli, Felice(F-SORBU-LJL)
Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori.
J. Math. Pures Appl. (9)157(2022), 243–281.

[6] Feola, Roberto(I-MILAN); Iandoli, Felice(F-SORBU-LJL)
Long time existence for fully nonlinear NLS with small Cauchy data on the circle.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)22(2021), no.1 109–182.


[5] Ivanovici, Oana(F-SORBU-LJL); Lebeau, Gilles(F-NICE-LD); Planchon, Fabrice(F-SORBU-IMJ)
Strichartz estimates for the wave equation on a 2D model convex domain.
J. Differential Equations300(2021), 830–880.

[4] Ivanovici, Oana(F-SORBU-LJL); Lebeau, Gilles(F-NICE-LD); Planchon, Fabrice(F-SORBU-IMJ)
New counterexamples to Strichartz estimates for the wave equation on a 2D model convex domain.
J. Éc. polytech. Math.8(2021) 1133–1157.

[3] Ivanovici, Oana (F-SORBU-LJL)
Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain.
Discrete Contin. Dyn. Syst.41(2021) no.12 5707–5742.


[2] Bernier, Joackim(F-NANT-LM); Feola, Roberto(F-NANT-LM); Grébert, Benoît(F-NANT-LM); Iandoli, Felice(F-SORBU-LJL)
Long-time existence for semi-linear beam equations on irrational Tori.
J. Dynam. Differential Equations33(2021), no.3 1363–1398.


[1] Planchon, Fabrice(F-NICE-LD); Tzvetkov, Nikolay(F-CEPO-LMA); Visciglia, Nicola(I-PISA)
Transport of Gaussian measures by the flow of the nonlinear Schrödinger equation.(English summary)
Math. Ann.378(2020) no.1-2 389–423.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Summary of the progress achieved:

- Parametrix construction for the wave equation in a general convex domain
- sharp dispersion estimates for the wave equation on a model convex domain
- dispersion estimates for Schrodinger equation inside convex domains and new applications to the cubic NLS
- long time behavior of nonlinear Schrodinger and KdV equations on compact manifolds
- sharp propagation of singularities inside convex domains
- dispersive estimates on general domains (no convexity assumption)
- sharp dispersive estimates for wave and Schrodinger equations outside a strictly convex obstacle in 3d ;
- counter-examples to dispersion in higher dimensions (at the Poisson-Arago spot)
- sharp dispersive and Strichartz bounds for the wave flow near a saddle point (where the curvature changes sign)
- relation between the exponential sum estimates and the dispersive bounds for the Schrodinger flow on the half line

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