Seminar on Analysis, Geometry and PDEs - Emanuele Haus (University of Roma Tre)

Tid: 2022-06-07 15:15 till 16:15 Seminarium

Normal form, existence time and dynamics of the Kirchhoff equation


In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with P. 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) of degree five 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 $\varepsilon^{-6}$.

In a recent work in progress with P. Baldi, F. Giuliani and M. Guardia, we use such effective terms of degree five to construct some special solutions exhibiting a chaotic-like behavior.

Om händelsen
Tid: 2022-06-07 15:15 till 16:15


eskil [dot] rydhe [at] math [dot] lu [dot] se

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Sidansvarig: webbansvarig@math.lu.se | 2017-05-23