Programme
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jeudi 6 juin 2024 ›
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09:00
10:00
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15:00
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19:00
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21:00
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›11:00 (1h)
Eliot Pacherie -------------------- Some examples of non unique minimizing travelling waves for a Schrödinger equation
We consider nonlinear Schrödinger equations with a nontrivial condition at infinity for a large class of nonlinearity. It has been shown by Mihai Maris that these equations admit travelling wave solutions for any subsonic speed, and they are constructed as the solutions of a minimizing problem. We are interested in the following question : do we always have uniqueness of the solution of this minimizing problem, up to the natural invariances of the problem ? In this talk, we will show how to construct a specific nonlinearity for which this is not the case. This is a current project in collaboration with Mihai Maris
›15:00 (1h)
Alberto Enciso ----------------------- Fluid equilibria with compact support and the fearful symmetry of Laplace eigenfunctions.
Schiffer’s conjecture in spectral geometry asserts that the only planar domains with a Neumann eigenfunction that is constant on the boundary are disks. If the hypothesis is relaxed so that the Neumann eigenfunction is only assumed to be locally constant on the boundary, we show that there are nontrivial doubly connected domains with this property, in spite of the fact that that the relaxed problem shares many rigidity properties with Schiffer. Furthermore, this has a nice application in fluid mechanics, as one can use these domains to construct nonradial Lipschitz continuous stationary solutions to the 2d incompressible Euler equations with compactly supported velocity.
›16:30 (1h)
Valeria Banica ----------------------- Multifractality in the limit evolution of polygonal vortex filaments.
With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann’s non-differentiable functions. These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. The result that I will present in this talk is that we determine for some of them the spectrum of singularities. The proofs rely on a careful design of Diophantine sets, which we study by using the Duffin-Schaeffer theorem and the Mass Transference Principle. This is a joint work with Daniel Eceizabarrena, Andrea Nahmod and Luis Vega.
›17:30 (1h)
Nicolas Camps ----------------------- Do Energy Cascade Mechanisms for NLS Persist on Generic Tori?
When the nonlinear Schr¨odinger equation (NLS) is posed on the square torus, exact resonances can lead to energy transfers and growth of higher-order Sobolev norms. In this presentation, we investigate whether these mechanisms persist under perturbations of the domain, when the exact resonances are replaced by quasi-resonances. Under generic Diophantine conditions on the torus, we prove that energy cascades are absent over longer time scales compared to those expected on the square torus. These results emphasize the importance of domain properties in the study of nonlinear waves. The proofs are based on normal form methods, and the analysis of the quasi-resonant interactions involves a frequency separation property. These are joint works with Joackim Bernier and Gigliola Staffilani.
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