Abstract
In order to prove our main result, the first step is to reduce the capillary-gravity water waves equations to a paradifferential system in a convenient complex unknown, suitable to prove energy estimates. The paralinearization of the water waves equations is obtained in Chap. 7, based on the analysis of Chap. 6. The goal of the present chapter is to present the general form of such a paradifferential system and to state the main Theorem concerning almost global existence of its solutions with a small initial datum. In a preliminary section we present the key algebraic properties of Reality, Parity and Reversibility of such a system that will play a fundamental role for getting energy estimates of the normal form system. The first part of the proof of the main Theorem is presented in the last section of the chapter.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Berti, M., Delort, JM. (2018). Complex Formulation of the Equation and Diagonalization of the Matrix Symbol. In: Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle. Lecture Notes of the Unione Matematica Italiana, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-99486-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-99486-4_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99485-7
Online ISBN: 978-3-319-99486-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)