Comptes Rendus
Partial Differential Equations
Unconditional well-posedness for subcritical NLS in Hs
[Unicité inconditionnelle pour l'équation de Schrödinger non-linéaire sous-critique dans Hs]
Comptes Rendus. Mathématique, Volume 345 (2007) no. 7, pp. 395-398.

On considère l'équation de Schrödinger linéaire sous-critique itu+Δu=|u|αu, sur Rn, n3, à donnée initiale u0 dans Hs(Rn). Si s1, Kato a démontré que si il existe une solution maximale, elle est unique dans C([0,Tmax),Hs). Les seuls résultats d'unicité connus auparavant étaient dans des sous-espaces stricts de cet espace. L'existence d'une solution étant connue pour s[0,1], l'équation de Schrödinger sous-critique est localement bien posée dans H1 sans condition supplémentaire pour l'unicité. Dans cette Note, nous généralisons le résultat d'unicité de Kato, montrant que l'équation est bien posée avec unicité inconditionnelle dans tous les espaces Hs, s[n2(n1),1].

Let n3 and consider the subcritical nonlinear Schrödinger equation, itu+Δu=|u|αu, with initial data u0Hs(Rn). When s1, Kato proved that if a maximal solution exists, then it is unique in C([0,Tmax),Hs). Previously, uniqueness had only been proven in strictly smaller subspaces. The existence of a solution is assured when s[0,1], so that the subcritical nonlinear Schrödinger equation is unconditionally locally well-posed in H1. We extend the uniqueness result so that the subcritical nonlinear Schrödinger equation is unconditionally locally well-posed in Hs when s[n2(n1),1].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.09.003

Keith M. Rogers 1

1 Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
@article{CRMATH_2007__345_7_395_0,
     author = {Keith M. Rogers},
     title = {Unconditional well-posedness for subcritical {NLS} in $ {H}^{s}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {395--398},
     publisher = {Elsevier},
     volume = {345},
     number = {7},
     year = {2007},
     doi = {10.1016/j.crma.2007.09.003},
     language = {en},
}
TY  - JOUR
AU  - Keith M. Rogers
TI  - Unconditional well-posedness for subcritical NLS in $ {H}^{s}$
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 395
EP  - 398
VL  - 345
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2007.09.003
LA  - en
ID  - CRMATH_2007__345_7_395_0
ER  - 
%0 Journal Article
%A Keith M. Rogers
%T Unconditional well-posedness for subcritical NLS in $ {H}^{s}$
%J Comptes Rendus. Mathématique
%D 2007
%P 395-398
%V 345
%N 7
%I Elsevier
%R 10.1016/j.crma.2007.09.003
%G en
%F CRMATH_2007__345_7_395_0
Keith M. Rogers. Unconditional well-posedness for subcritical NLS in $ {H}^{s}$. Comptes Rendus. Mathématique, Volume 345 (2007) no. 7, pp. 395-398. doi : 10.1016/j.crma.2007.09.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.003/

[1] T. Cazenave; F.B. Weissler The Cauchy problem for the nonlinear Schrödinger equation in H1, Manuscripta Math., Volume 61 (1988) no. 4, pp. 477-494

[2] T. Cazenave; F.B. Weissler The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear Anal., Volume 14 (1990) no. 10, pp. 807-836

[3] T. Cazenave; F.B. Weissler Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 75-100

[4] G. Furioli; F. Planchon; E. Terraneo Unconditional well-posedness for semilinear Schrödinger and wave equations in Hs, Harmonic Analysis at Mount Holyoke, Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 147-156

[5] G. Furioli; E. Terraneo Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equation in H˙s(Rn), Commun. Contemp. Math., Volume 5 (2003) no. 3, pp. 349-367

[6] J. Ginibre; G. Velo The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 2 (1985) no. 4, pp. 309-327

[7] T. Kato On nonlinear Schrödinger equations. II. Hs-solutions and unconditional well-posedness, J. Anal. Math., Volume 67 (1995), pp. 281-306

[8] M. Keel; T. Tao Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980

[9] R.S. Strichartz Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Volume 44 (1977) no. 3, pp. 705-714

[10] M.C. Vilela Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., Volume 359 (2007) no. 5, pp. 2123-2136 (electronic)

[11] K. Yajima Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., Volume 110 (1987) no. 3, pp. 415-426

Cité par Sources :

Commentaires - Politique