• Tingueu el micròfon apagat tota l'estona excepte durant el "Cafè i galetes".
• Podeu tenir la càmera encesa si voleu, però en cas de saturaciò de línies, caldrà que la tanqueu.
• Si voleu fer una pregunta, si us plau demaneu torn a través del xat i espereu que els moderadors us donin la paraula.
• Si és possible, entreu amb antelació. Els que no useu un compte de correu de la UPC haureu d'esperar a ser admesos per accedir-hi. (Això aplica a les sessions Google Meet de la UPC).

Dia: Dimecres, 25 de maig de 2022

Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

També ONLINE https://meet.google.com/zvg-pajn-owr

• Hora: 15h00m

A càrrec de: Pietro Baldi (Università di Napoli)

Títol: Normal form and existence time for the Kirchhoff equation

Resum: We consider the Kirchhoff equation ∂_ttu − Δu (1 + ∫_𝕋^d |∇u|²dx) = 0 on the d-dimensional torus 𝕋^d. This is a quasi-linear PDE with the structure of an infinite-dimensional Hamiltonian system, originally proposed as a nonlinear model for the oscillations of elastic strings and membranes.

In the talk we present two recent results, obtained in collaboration with Emanuele Haus, about the Cauchy problem with initial data of size ε in Sobolev class.

In the first result we make a first step of normal form, preceded by a nonlinear transformation that diagonalizes the operator at the highest order; this preparatory transformation is required by the quasi-linear nature of the problem. After the normal form step, the resonant cubic terms in the transformed equation give no contribution to the energy estimates. As a consequence, we obtain a lower bound ε^-4 for the lifespan of the solutions (the standard local theory gives ε^-2).
In the second result, we begin with performing a second step of normal form, and we obtain an effective equation for the dynamics of the system at the quintic order, which contain resonances corresponding to nontrivial terms in the energy estimates. We introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound ε^-6 for the lifespan of the corresponding solutions. The mechanism at the base of this improvement is an averaging effect; under these nonresonance conditions, the growth rate of the “superactions” of the effective equations on large time intervals is smaller than its a priori estimate based on the normal form for the cubic terms.
• Hora: 16h15m

A càrrec de: Alberto Pérez-Cervera (Universidad Complutense de Madrid)

Títol: Isostables for Stochastic Oscillators

Resum: Phase-Amplitude variables are indispensable tools to characterize oscillatory dynamics. However, achieving an extension of these tools to stochastic oscillators, i.e. noisy excitable systems, has remained an open question until very recently. In this talk, we will present a framework (inspired on the parameterisation method) for the ‘phase-amplitude’ description of stochastic oscillators [1,2], and discuss possible applications.

This is a joint work with B.Lindner, P.Thomas, P.Houzelstein and B.Gutkin

1. B.Lindner, P.Thomas. 'Asymptotic Phase for Stochastic Oscillators'. Phys Rev Lett 2014
2. A.Pérez-Cervera, B.Lindner, P.Thomas. 'Isostables for Stochastic Oscillators'. Phys Rev Lett 2021

Last updated: Fri May 20 22:24:48 2022