Existence of global solutions and global attractor for the third order Lugiato–Lefever equation on <strong>T</strong>

Miyaji, Tomoyuki; Tsutsumi, Yoshio

doi:10.1016/j.anihpc.2016.12.004

Existence of global solutions and global attractor for the third order Lugiato–Lefever equation on T

Miyaji, Tomoyuki ; Tsutsumi, Yoshio

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1707-1725

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Résumé

We show the existence of global solution and the global attractor in $L^{2} (T)$ for the third order Lugiato–Lefever equation on T. Without damping and forcing terms, it has three conserved quantities, that is, the $L^{2} (T)$ norm, the momentum and the energy, but the leading term of the energy functional is not positive definite. So only the $L^{2}$ norm conservation is useful for the third order Lugiato–Lefever equation unlike the KdV and the cubic NLS equations. Therefore, it seems important and natural to construct the global attractor in $L^{2} (T)$ . For the proof of the global attractor, we use the smoothing effect of cubic nonlinearity for the reduced equation.

MR Zbl

DOI : 10.1016/j.anihpc.2016.12.004

Classification : 35Q55, 35Q53, 35A01, 35B41
Keywords: Third order Lugiato–Lefever equation, Strichartz' estimate on one dimensional torus, Nonlinear smoothing effect, Global attractor

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     author = {Miyaji, Tomoyuki and Tsutsumi, Yoshio},
     title = {Existence of global solutions and global attractor for the third order {Lugiato{\textendash}Lefever} equation on {<strong>T</strong>}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1707--1725},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
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     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.12.004/}
}

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Miyaji, Tomoyuki; Tsutsumi, Yoshio. Existence of global solutions and global attractor for the third order Lugiato–Lefever equation on T. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1707-1725. doi: 10.1016/j.anihpc.2016.12.004

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