The third order Benjamin-Ono equation on the torus: Well-posedness, traveling waves and stability
Gassot, Louise1, 2
1 a Département de Mathématiques et Applications, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France
2 b Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France
Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 815-840.
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We consider the third order Benjamin-Ono equation on the torus

∂tu=∂x(−∂xxu−32uH∂xu−32H(u∂xu)+u3).
We prove that for any t∈R, the flow map continuously extends to Hr,0s(T) if s≥0, but does not admit a continuous extension to Hr,0−s(T) if 0<s<12. Moreover, we show that the extension is weakly sequentially continuous in Hr,0s(T) if s>0, but is not weakly sequentially continuous in Lr,02(T). We then classify the traveling wave solutions for the third order Benjamin-Ono equation in Lr,02(T) and study their orbital stability.

Reçu le :
Révisé le :
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DOI : 10.1016/j.anihpc.2020.09.004
Classification : 37K10, 35Q53
Mots-clés : Benjamin-Ono equation, Benjamin-Ono hierarchy, Birkhoff coordinates, Well-posedness, Traveling waves, Orbital stability
Gassot, Louise 1, 2

1 a Département de Mathématiques et Applications, École Normale Supérieure, CNRS, PSL University, 75005 Paris, France
2 b Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d'Orsay, 91405 Orsay, France 
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     title = {The third order {Benjamin-Ono} equation on the torus: {Well-posedness,} traveling waves and stability},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {815--840},
     publisher = {Elsevier},
     volume = {38},
     number = {3},
     year = {2021},
     doi = {10.1016/j.anihpc.2020.09.004},
     mrnumber = {4227053},
     zbl = {1467.37061},
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Gassot, Louise. The third order Benjamin-Ono equation on the torus: Well-posedness, traveling waves and stability. Annales de l'I.H.P. Analyse non linéaire, mai – juin 2021, Tome 38 (2021) no. 3, pp. 815-840. doi : 10.1016/j.anihpc.2020.09.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.09.004/

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