Comptes Rendus
Partial differential equations
A remark on global well-posedness of the derivative nonlinear Schrödinger equation on the circle
[Une remarque sur le caractère globalement bien posé de l'équation de Schrödinger non linéaire avec dérivée sur le cercle]
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 837-841.

On considère dans cette note l'équation de Schrödinger avec dérivée sur le cercle. En particulier, en adaptant l'argument récent de Wu au cas periodique, on prouve que cette équation est globalement bien posée dans H1(T), pourvu que la masse soit inférieure à 4π. En outre, ce seuil pour la masse est indépendant des périodes spatiales.

In this note, we consider the derivative nonlinear Schrödinger equation on the circle. In particular, by adapting Wu's recent argument to the periodic setting, we prove its global well-posedness in H1(T), provided that the mass is less than 4π. Moreover, this mass threshold is independent of spatial periods.

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DOI : 10.1016/j.crma.2015.06.015

Razvan Mosincat 1, 2 ; Tadahiro Oh 1, 2

1 School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
2 The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
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Razvan Mosincat; Tadahiro Oh. A remark on global well-posedness of the derivative nonlinear Schrödinger equation on the circle. Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 837-841. doi : 10.1016/j.crma.2015.06.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.06.015/

[1] M. Agueh Sharp Gagliardo–Nirenberg inequalities and mass transport theory, J. Dyn. Differ. Equ., Volume 18 (2006) no. 4, pp. 1069-1093

[2] J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., Volume 33 (2001) no. 3, pp. 649-669

[3] J. Colliander; M. Keel; G. Staffilani; H. Takaoka; T. Tao A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., Volume 34 (2002) no. 1, pp. 64-86

[4] N. Hayashi; T. Ozawa Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., Volume 25 (1994) no. 6, pp. 1488-1503

[5] S. Herr On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not. (2006) (Art. ID 96763, 33 p)

[6] J. Lebowitz; H. Rose; E. Speer Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., Volume 50 (1988) no. 3–4, pp. 657-687

[7] A. Nahmod; T. Oh; L. Rey-Bellet; G. Staffilani Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., Volume 14 (2012), pp. 1275-1330

[8] S.B. Tan Blow-up solutions for mixed nonlinear Schrödinger equations, Acta Math. Sin. Engl. Ser., Volume 20 (2004) no. 1, pp. 115-124

[9] M. Weinstein Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., Volume 87 (1982–1983) no. 4, pp. 567-576

[10] Y.Y.S. Win Global well-posedness of the derivative nonlinear Schrödinger equations on T, Funkc. Ekvacioj, Volume 53 (2010) no. 1, pp. 51-88

[11] Y. Wu Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space, Anal. PDE, Volume 6 (2013) no. 8, pp. 1989-2002

[12] Y. Wu Global well-posedness on the derivative nonlinear Schrödinger equation revisited | arXiv

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