Normal form and existence time for the Kirchhoff equation

Abstract: In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with Pietro Baldi. Computing the first step of quasilinear normal form, we erase from the equation all the cubic terms giving nonzero contribution to the energy estimates; thus we deduce that, for small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ coming from the linear theory). In the second step of normal form, there remain some resonant terms (which cannot be erased) that give a non-trivial contribution to the energy estimates; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$.

mathematical physicsclassical analysis and ODEsdynamical systemsnumerical analysisprobabilitysymplectic geometry

Audience: researchers in the topic

Third DinAmicI Day

Series comments: The Giornata DinAmica (DAI Day) of the DinAmicI, the Community of Italian Dynamicists, takes place every two years and includes, in addition to scientific seminars, the assembly of members.

One of the aims of the DinAmicI is to promote young researchers: at least half of the invited speakers are in the early stage of their careers.

This third edition of the Day is exceptionally held online due to the restrictions caused by Covid and, exceptionally, the talks will be distributed over two afternoons.