×
Cazenave, Thierry; Weissler, Fred B.

The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^ s\). (English) Zbl 0706.35127

Nonlinear Anal., Theory Methods Appl. 14, No. 10, 807-836 (1990).
Consider the Cauchy problem for the nonlinear Schrödinger equation in \({\mathbb{R}}^ N\), \[ (NLS)\quad iu_ t+\Delta u+\lambda | u|^{\alpha}u=0,\quad u(0)=\phi. \] Here, \(\phi\) is a given initial value, \(\lambda\) is a real parameter and \(\alpha >0\) is arbitrary (up to technical conditions that intervene only for dimensions \(N\geq 7)\). It is established that if \(0\leq s<N/2\) and \(\alpha\leq 4/(N-2s)\), then problem (NLS) is locally well posed in the Sobolev space \(H^ s({\mathbb{R}}^ N)\). In the case \(\alpha =4/(N-2s)\), if \(\| (-\Delta)^{s/2}\phi \|_{L^ 2}\) is sufficiently small, then the solution is global and decays as \(t\to \infty\) at the same rate as the free solution. The method relies on estimates for \(e^{it\Delta}\) in certain Besov spaces, on estimates of \(| u|^{\alpha}u\) in the same Besov spaces, and on a fixed point argument.
Reviewer: Th.Cazenave

Cited in 2 Reviews
Cited in 411 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs

Keywords:

global existence; nonlinear Schrödinger equation; Besov spaces
Cite Review PDF
Full Text: DOI

References:

[1] Bergh, J.; Löfstöm, J., Interpolation Spaces (1976), Springer: Springer New York · Zbl 0344.46071
[2] Cazenave, T.; Weissler, F. B., The Cauchy problem for the nonlinear Schrödinger equation in \(H^1\), Manuscripta math., 61, 477-494 (1988) · Zbl 0696.35153
[3] Cazenave, T.; Weissler, F. B., Some remarks on the nonlinear Schrödinger equation in the critical case, (Gill, T. L.; Zachary, W. W., Nonlinear Semigroups, Partial Differential Equations, and Attractors, 1394 (1989), Springer: Springer Berlin), 18-29, Lecture Notes in Mathematics · Zbl 0694.35170
[4] Fabes, E. B.; Lewis, J. E.; Riviere, N. M., Boundary value problems for the Navier-Strokes equations, Am. J. Math., 99, 626-668 (1977) · Zbl 0386.35037
[5] Fujita, H.; Kato, T., On the Navier-Strokes initial value problem I, Archs. ration Mech. Analysis, 16, 269-315 (1964) · Zbl 0126.42301
[6] Giga, Y., Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions for the Navier-Stokes system, J. diff. Eqns, 62, 168-212 (1986) · Zbl 0577.35058
[7] Giga, Y.; Kohn, R. V., Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36, 1-40 (1987) · Zbl 0601.35052
[8] Giga, Y.; Kohn, R. V.; Papanicolaou, G., Removability of blowup points for similinear heat equations, Proc. EQUADIFF conf. (August 1987)
[9] Giga, Y.; Miyakawa, T., Solutions in \(L_r\) of the Navier-Stokes initial value problem, Archs. ration. Mech. Analysis, 89, 267-281 (1985) · Zbl 0587.35078
[10] Ginibre, J.; Velo, G., Sur une équation de Schrödinger non linéaire avec interaction non locate, (Brezis, H.; Lions, J. L., Nonlinear partial differential equations and their applications. Nonlinear partial differential equations and their applications, Collége de France Seminar (1982), Pitman: Pitman Boston), 155-199, Collége de France Seminar · Zbl 0497.35024
[11] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Analyse Non Linéaire, 2, 309-327 (1985) · Zbl 0586.35042
[12] Ginibre, J.; Velo, G., The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189, 487-505 (1985) · Zbl 0549.35108
[13] Haraux, A.; Weissler, F. B., Non-uniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31, 167-189 (1982) · Zbl 0465.35049
[14] Kato, T., Strong \(L^p\)-solutions of the Navier-Stokes equations in \(\textbf{R}^m\) with applications to weak solutions, Math. Z., 187, 471-480 (1984) · Zbl 0545.35073
[15] Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Théorique, 46, 113-129 (1987) · Zbl 0632.35038
[16] Kato, T.; Fujita, H., On the nonstationary Navier-Stokes system, Rc. Semin. Mat. Univ. Padova, 32, 243-260 (1962) · Zbl 0114.05002
[17] Merle, F., Limit of the solution of the nonlinear Schödinger equation at the blow-up time, J. funct. Analysis, 84, 201-214 (1989) · Zbl 0681.35078
[18] Miyakawa, T., On the initial value problem for the Navier-Stokes equations in \(L^p\) spaces, Hiroshima Math. J., 11, 9-20 (1981) · Zbl 0457.35073
[19] Mueller, C. E.; Weissler, F. B., Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J., 34, 881-913 (1985) · Zbl 0597.35057
[20] Reed, M., Abstract Nonlinear Wave Equations, Lecture Notes in Mathematics, 507 (1976), Springer: Springer New York · Zbl 0317.35002
[21] Simon, J., Regularité de la composition de deux fonctions et applications, Boll. Un. mat. Ital., 16B, 5, 501-522 (1979) · Zbl 0409.35076
[22] Sobolevskii, P. E., Study of Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces, Soviet Math. Dokl., 5, 720-723 (1964)
[23] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501
[24] Strauss, W. A., Nonlinear invariant wave equations, (Velo, G.; Wightman, A. S., Lecture Notes in Physics. Lecture Notes in Physics, Invariant Wave Equations, 73 (1978), Springer: Springer New York), 197-249
[25] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., 44, 705-714 (1977) · Zbl 0372.35001
[26] Tsutsumi, Y., \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, 30, 115-125 (1987) · Zbl 0638.35021
[28] von Wahl, W., Regularity questions for the Navier-Stokes equations, (Rautman, R., Lecture Notes in Mathematics. Lecture Notes in Mathematics, Approximation Methods for Navier-Stokes Problems, 771 (1980), Springer: Springer New York), 538-542 · Zbl 0451.35051
[29] von Wahl, W., The Equations of Navier-Stokes and Abstract Parabolic Equations (1985), Vieweg: Vieweg Wiesbaden · Zbl 1409.35168
[30] von Wahl, W., Global solutions to evolutions equations of parabolic type, (Favini, A.; Obrecht, E., Lecture Notes in Mathematics. Lecture Notes in Mathematics, Differential Equations in Banach Spaces, 1223 (1986), Springer: Springer New York), 254-266 · Zbl 0624.34068
[31] Weinstein, M., On the structure and formation of singularities of solutions to nonlinear dispersive equations, Communs partial diff. Eqns, 11, 545-565 (1986) · Zbl 0596.35022
[32] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. math J., 29, 79-102 (1980) · Zbl 0443.35034
[33] Weissler, F. B., The Navier-Stokes initial value problem in \(L^p\), Archs ration. Mech. Analaysis, 74, 219-230 (1980) · Zbl 0454.35072
[34] Weissler, F. B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38, 29-40 (1981) · Zbl 0476.35043
[35] Weissler, F. B., \(L^p\)-energy and blow-up for a semilinear heat equation, (Proc. of Symp. in Pure Mathematics, 45 (1986), Am. Math. Soc: Am. Math. Soc Providence), 545-551, Part 2 · Zbl 0631.35049
[36] Yajima, K., Existence of solutions for Schrödinger evolutions equations, Communs math. Phys., 110, 415-426 (1987) · Zbl 0638.35036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.