Continuous dependence for NLS in fractional order spaces

Cazenave, Thierry ; Fang, Daoyuan ; Han, Zheng

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 135-147 / Harvested from Numdam

Résumé

For the nonlinear Schrödinger equation $ i{u}_{t}+\mathrm{\Delta }u+\lambda {|u|}^{\alpha }u=0$ in $ {\mathbb{R}}^{N}$, local existence of solutions in $ {H}^{s}$ is well known in the $ {H}^{s}$-subcritical and critical cases $ 0<\alpha ⩽4/(N-2s)$, where $ 0<s<\mathrm{min}\{N/2,1\}$. However, even though the solution is constructed by a fixed-point technique, continuous dependence in $ {H}^{s}$ does not follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous $ {H}^{s}\to {H}^{s}$. If, in addition, $ \alpha ⩾1$ then the flow is locally Lipschitz.

Détail

Publié le : 2011-01-01

MR 2765515 | Zbl 1209.35124

DOI : https://doi.org/10.1016/j.anihpc.2010.11.005
Classification:  35Q55,  35B30,  46E35

Bibliographie

[1] J. Bergh, J. Löfström, Interpolation Spaces, Springer, New York (1976) | Zbl 0128.35104

[2] G. Bourdaud, M. Lanza De Cristoforis, Regularity of the symbolic calculus in Besov algebras, Studia Math. 184 no. 3 (2008), 271-298 | MR 2369144 | Zbl 1139.46030

[3] G. Bourdaud, M. Lanza De Cristoforis, W. Sickel, Superposition operators and functions of bounded p-variation, Rev. Mat. Iberoamericana 22 no. 2 (2006), 455-487 | MR 2294787 | Zbl 1134.46015

[4] G. Bourdaud, M. Lanza De Cristoforis, W. Sickel, Superposition operators and functions of bounded p-variation. II, Nonlinear Anal. 62 no. 3 (2005), 483-517 | MR 2147980 | Zbl 1090.47050

[5] G. Bourdaud, M. Moussai, W. Sickel, Composition operators on Lizorkin–Triebel spaces, preprint, 2009. | MR 2652183

[6] H. Brezis, P. Mironescu, Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 no. 4 (2001), 387-404 | MR 1877265 | Zbl 1023.46031

[7] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10, New York University, Courant Institute of Mathematical Sciences/Amer. Math. Soc., New York/Providence, RI (2003) | MR 2002047 | Zbl 1055.35003

[8] T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $ {H}^{s}$, Nonlinear Anal. 14 no. 10 (1990), 807-836 | MR 1055532 | Zbl 0706.35127

[9] T. Cazenave, F.B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 no. 1 (1992), 75-100 | MR 1171761 | Zbl 0763.35085

[10] D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 no. 1 (2005), 1-24 | MR 2134950 | Zbl 1071.35025

[11] G. Furioli, E. Terraneo, Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equation, Commun. Contemp. Math. 5 no. 3 (2003), 349-367 | MR 1992354 | Zbl 1050.35102

[12] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 no. 1 (1987), 113-129 | Numdam | MR 877998 | Zbl 0632.35038

[13] T. Kato, On nonlinear Schrödinger equations, II. $ {H}^{s}$-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281-306 | MR 1383498 | Zbl 0848.35124

[14] T. Kato, An $ {L}^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and Scattering Theory and Applications, Adv. Stud. Pure Math. vol. 23 (1994), 223-238 | MR 1275405 | Zbl 0832.35131

[15] M. Keel, T. Tao, Endpoint Strichartz inequalities, Amer. J. Math. 120 (1998), 955-980 | MR 1646048 | Zbl 0922.35028

[16] C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 no. 3 (2006), 645-675 | Zbl 1115.35125

[17] R. Killip, M. Visan, Nonlinear Schrödinger Equations at Critical Regularity, Clay Math. Proc. vol. 10 (2009) | MR 3098643

[18] H. Pecher, Solutions of semilinear Schrödinger equations in $ {H}^{s}$, Ann. Inst. H. Poincaré Phys. Théor. 67 no. 3 (1997), 259-296 | MR 1472820 | Zbl 0888.35101

[19] K.M. Rogers, Unconditional well-posedness for subcritical NLS in $ {H}^{s}$, C. R. Math. Acad. Sci. Paris 345 no. 7 (2007), 395-398 | MR 2361505 | Zbl 1156.35094

[20] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter Ser. Nonlinear Anal. Appl. vol. 3, Walter de Gruyter & Co., Berlin (1996) | MR 1419319 | Zbl 0873.35001

[21] M. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714 | MR 512086 | Zbl 0372.35001

[22] T. Tao, M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 118 (2005) | MR 2174550 | Zbl 1245.35122

[23] H. Triebel, Theory of Function Spaces, Monogr. Math. vol. 78, Birkhäuser Verlag, Basel (1983) | MR 781540

[24] Y. Tsutsumi, $ {L}^{2}$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 no. 1 (1987), 115-125 | MR 915266 | Zbl 0638.35021

[25] M.-C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), 2123-2136 | MR 2276614 | Zbl 1196.35074

[26] Y.Y.S. Win, Y. Tsutsumi, Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation, Hokkaido Math. J. 37 no. 4 (2008), 839-859 | MR 2474179 | Zbl 1173.35696

[27] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), 415-426 | MR 891945 | Zbl 0638.35036