scholarly journals Extended lifespan of the fractional BBM equation

2021 ◽  
pp. 1-21
Author(s):  
Dag Nilsson
Keyword(s):  
Normal Form ◽  
Initial Data ◽  
Energy Method ◽  
Existence Time ◽  
Extended Lifespan ◽  
Form Transformation ◽  
Normal Form Transformation ◽  
Bbm Equation

For 0 < α < 1, N ⩾ 2 and with initial data ‖ u 0 ‖ H N + α 2 = ε, sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂ t u + ∂ x u + u ∂ x u + | D | α ∂ t u = 0, can be extended from the hyperbolic existence time 1 ε to 1 ε 2 . For the proof we use a quasilinear modified energy method, based on a normal form transformation as in Hunter, Ifrim, Tataru, Wong (Proc. Am. Math. Soc., 143(8) (2015) 3407–3412).

scholarly journals On the Normal Form of the Kirchhoff Equation

2020 ◽  
Author(s):  
Pietro Baldi ◽  
Emanuele Haus
Keyword(s):  
Normal Form ◽  
Negative Answer ◽  
Kirchhoff Equation ◽  
Energy Estimates ◽  
Second Step ◽  
Dimensional Torus ◽  
Form Transformation ◽  
Normal Form Transformation ◽  
Nonzero Contribution ◽  
Formal Calculation

Abstract Consider the Kirchhoff equation $$\begin{aligned} \partial _{tt} u - \Delta u \Big ( 1 + \int _{\mathbb {T}^d} |\nabla u|^2 \Big ) = 0 \end{aligned}$$ ∂ tt u - Δ u ( 1 + ∫ T d | ∇ u | 2 ) = 0 on the d-dimensional torus $$\mathbb {T}^d$$ T d . In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

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