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
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 + ∫𝕋d |∇u|2dx) = 0 on the d-dimensional torus 𝕋d. 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.
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
Last updated: Fri May 20 22:24:48 2022