Long-Time Existence for Semi-linear Beam Equations on Irrational Tori
Por:
Bernier J., Feola R., Grébert B., Iandoli F.
Publicada:
1 ene 2021
Categoría:
Analysis
Resumen:
We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n- 1 with n= 3 and d= 2. If e« 1 is the size of the initial datum, we prove that the lifespan Te of solutions is O(e-A(n-2)-) where A=A(d,n)=1+3d-1 when n is even and A=1+3d-1+max(4-dd-1,0) when n is odd. For instance for d= 2 and n= 3 (quadratic nonlinearity) we obtain Te=O(e-6-), much better than O(e- 1) , the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of 2+1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
Filiaciones:
Bernier J.:
Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, Nantes Cedex 03, 44322, France
Feola R.:
Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, Nantes Cedex 03, 44322, France
Grébert B.:
Laboratoire de Mathématiques Jean Leray, Université de Nantes, UMR CNRS 6629, 2, rue de la Houssinière, Nantes Cedex 03, 44322, France
Iandoli F.:
Laboratoire Jacques-Louis Lions, Sorbonne Université, UMR CNRS 7598, 4, Place Jussieu, Paris Cedex 05, 75005, France
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