Small solutions to nonlinear Schrödinger equations. (English) Zbl 0786.35121

A local existence theorem is proved for solutions of the Cauchy problem for the nonlinear Schrödinger equation of the following form: \[ iu_ t=-\Delta u+iP(u,\nabla u,\overline u,\nabla \overline u), \] where \(u\) is a complex function of time and \(n\) spatial variables, and \(P\) is a polynomial function (without constant or linear terms) of a degree \(s\). The existence of solutions at finite values of time is proved, using assumptions which establish boundedness of the initial data.
The main technical ingredient in proving the theorem is the so-called smoothing effect of Kato type. The theorem is proved separately for four different cases corresponding, respectively, to \(n=1\) and \(n \geq 2\), and \(s=2\) and \(s \geq 3\). In each case, different specific boundedness conditions should be imposed on the initial data in order to prove the local existence of the solution.


35Q55 NLS equations (nonlinear Schrödinger equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs
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[1] Carbery, A., Radial Fourier multipliers and associated maximal function, North Holland Math. Studies, III, 49-55 (1985)
[3] Cazenave, T.; Weissler, F. B., Some remarks on the nonlinear Schrödinger equation in the critical case, Lecture in Math, Vol. 1392, 18-29 (1989), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0694.35170
[4] Christ, F. M.; Weinstein, M., Dispersive small amplitude solution to the generalized Korteweg-de Vries equation, J. Funct. Anal., Vol. 100, 87-109 (1991) · Zbl 0743.35067
[5] Coifman, R. R.; Meyer, Y., Au-delà des opérateurs pseudo-différentiel, Astérisque, Vol. 57 (1973)
[6] Constantin, P.; Saut, J.-C., Local smoothing properties of dispersive equations, J. Amer. Math., Soc., Vol. 1, 413-446 (1989) · Zbl 0667.35061
[7] Dahlberg, B.; Kenig, C. E., A note an almost every where behavior of solutions to the schrödinger equations, Lecture Notes in Math., Vol. 908, 205-208 (1982), Springer-Verlag: Springer-Verlag Berlin, New York
[8] Ginibre, J.; Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equation, J. Math, pures et appl., Vol. 64, 363-401 (1985) · Zbl 0535.35069
[9] Ginibre, J.; Velo, G., On a class of Schrödinger equations, J. Funct. Anal., Vol. 32, 1-71 (1979) · Zbl 0396.35029
[10] Ginibre, J.; Tsutsumi, Y., Uniqueness for the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., Vol. 20, 1388-1425 (1989) · Zbl 0702.35224
[11] Ghidaglia, J. M.; Saut, J.-C., On the initial value problem for the Davey-Stewarson systems, Nonlinearity, Vol. 3, 475-506 (1990) · Zbl 0727.35111
[12] Glassey, R. T., On the blowing up solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., Vol. 18, 1794-1797 (1979) · Zbl 0372.35009
[13] Hayashi, N., Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J, Vol. 62, 575-592 (1991)
[14] Hayashi, N.; Nakamitsu, K.; Tsutsumi, M., On solutions to the initial value problem for the nonlinear Schrödinger equations in one dimensions, Math. Z., Vol. 192, 637-650 (1986) · Zbl 0617.35025
[15] Hayashi, N.; Nakamitsu, K.; Tsutsumi, M., On solutions to the initial value problem for the nonlinear Schrödinger equations, J. Funct. Anal., Vol. 71, 218-245 (1987) · Zbl 0657.35033
[16] Hayashi, N.; Saitoh, S., Analyticity and global existence of small solutions to some nonlinear Shrödinger equations, Comm. Math. Phys., Vol. 129, 27-41 (1990) · Zbl 0705.35132
[18] Kato, T., Nonlinear Schrödinger equation, Schrodinger operators, (Holden, H.; Jensen, A., Lecture Notes in Physics, Vol. 345 (1989), Springer-Verlag: Springer-Verlag Berlin, New York), 218-263 · Zbl 0698.35131
[19] Kato, T., On the Cauchy problem for the (generalized) Kortewed-de Vries equation, Advances in Math. Supp. Studies, Studies in Applied Math., Vol. 8, 93-128 (1983) · Zbl 0549.34001
[20] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., Vol. 41, 891-907 (1988) · Zbl 0671.35066
[21] Kaup, D. J.; Newell, A. C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., Vol. 19, 798-801 (1978) · Zbl 0383.35015
[22] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., Vol. 4, 323-347 (1991) · Zbl 0737.35102
[23] Kenig, C. E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana University Math. J., Vol. 40, 33-69 (1991) · Zbl 0738.35022
[25] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for generalized Korteweg-de Vries via contraction principle, Comm. Pure Appl. Math., Vol. 46, 527-620 (1993) · Zbl 0808.35128
[26] Kenig, C. E.; Ruiz, A., A strong type (2, 2) estimate for the maximal function associated to the Schrödinger equation, Trans. Amer. Math. Soc., Vol. 280, 239-246 (1983) · Zbl 0525.42011
[27] Klainerman, S., Long time behavior of solutions to nonlinear evolutions equations, Arch. Ration. Mech. and Analysis, 78, 73-98 (1981) · Zbl 0502.35015
[28] Klainerman, S.; Ponce, G., Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., Vol. 36, 133-141 (1983) · Zbl 0509.35009
[29] Shatah, J., Global existence of small solutions to nonlinear evolution equations, J. Diff. Eqs., Vol. 46, 409-423 (1982) · Zbl 0518.35046
[30] Simon, J.; Taflin, E., Wave operators and analytic solutions for systems of systems of nonlinear Klein-Gordon equations and of non-linear Schrödinger equations, Comm. Math. Phys., 99, 541-562 (1985) · Zbl 0615.47034
[31] Sjölin, P., Regularity of solutions to the Schrödinger equations, Duke Math., 55, 699-715 (1987) · Zbl 0631.42010
[32] Stein, E. M., Oscillaroty integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, 307-355 (1986), Princeton University Press · Zbl 0618.42006
[33] Stein, E. M.; Weiss, G., Introduction to Fourier Analysis in Eucliden Spaces (1971), Princeton University Press · Zbl 0232.42007
[34] Srauss, W. A., Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 110-133 (1981) · Zbl 0466.47006
[35] Strichartz, R. S., Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Vol. 44, 705-714 (1977) · Zbl 0372.35001
[36] Tsutsumi, Y., Global strong solutions for nonlinear Schrödinger equation, Nonlinear Anal., 11, 1143-1154 (1987) · Zbl 0657.35032
[37] Tsutsumi, Y., \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, Vol. 31, 115-125 (1987) · Zbl 0638.35021
[38] Tsutsumi, M.; Fukuda, I., On solutions of the derivative nonlinear Schrödinger equation. Existence and Uniqueness Theorem, Funkcialaj Ekvacioj, 23, 259-277 (1980) · Zbl 0478.35032
[39] Tsutsumi, M.; Fukuda, I., On solutions of the derivative nonlinear Schröndinger equation. II, Funkcialaj Ekvacioj, 24, 85-94 (1981) · Zbl 0491.35016
[40] Vega, L., Doctoral Thesis (1987), Universidad Autonoma de Madrid: Universidad Autonoma de Madrid Spain
[41] Vega, L., The Schrödinger eqution: pointwise convergence to the initial date, Proc. Amer. Math. Soc., Vol. 102, 874-878 (1988) · Zbl 0654.42014
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