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) no. 1, pp. 135-147.

For the nonlinear Schrödinger equation $i{u}_t+\Delta u+\lambda {\left|u\right|}^{\alpha }u=0$ in ${ℝ}^N$, local existence of solutions in $H^s$ is well known in the $H^s$-subcritical and critical cases $0<\alpha ⩽4/(N-2s)$, where $0<s<\min \{N/2,1\}$. 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.

Classification : 35Q55, 35B30, 46E35
Mots clés : Schrödinger's equation, Initial value problem, Continuous dependence, Fractional order Sobolev spaces, Besov spaces

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

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

[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 | EuDML | MR | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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