On the role of quadratic oscillations in nonlinear Schrödinger equations II. The 𝐿²-critical case

Carles, Rémi; Keraani, Sahbi

doi:10.1090/S0002-9947-06-03955-9

On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case
HTML articles powered by AMS MathViewer

by Rémi Carles and Sahbi Keraani PDF

Trans. Amer. Math. Soc. 359 (2007), 33-62 Request permission

Abstract:

We consider a nonlinear semi-classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian–Kammerer and I. Gallagher for $L^2$-supercritical power-like nonlinearities and more general initial data. The present results concern the $L^2$-critical case, in space dimensions $1$ and $2$; we describe the set of non-linearizable data, which is larger, due to the scaling. As an application, we make precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrödinger equation. The proof relies on linear and nonlinear profile decompositions.

References

Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. MR 1705001
J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
Jean Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 83–112. MR 1315543
J. Bourgain, Refinements of Strichartz’ inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices 5 (1998), 253–283. MR 1616917, DOI 10.1155/S1073792898000191
Jean Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197–215 (1998). Dedicated to Ennio De Giorgi. MR 1655515
Rémi Carles, Geometric optics with caustic crossing for some nonlinear Schrödinger equations, Indiana Univ. Math. J. 49 (2000), no. 2, 475–551. MR 1793681, DOI 10.1512/iumj.2000.49.1804
Rémi Carles, Clotilde Fermanian Kammerer, and Isabelle Gallagher, On the role of quadratic oscillations in nonlinear Schrödinger equations, J. Funct. Anal. 203 (2003), no. 2, 453–493. MR 2003356, DOI 10.1016/S0022-1236(03)00212-X
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
Thierry Cazenave and Fred B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987) Lecture Notes in Math., vol. 1394, Springer, Berlin, 1989, pp. 18–29. MR 1021011, DOI 10.1007/BFb0086749
Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. MR 1055532, DOI 10.1016/0362-546X(90)90023-A
Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
Isabelle Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France 129 (2001), no. 2, 285–316 (English, with English and French summaries). MR 1871299, DOI 10.24033/bsmf.2398
Patrick Gérard, Description du défaut de compacité de l’injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213–233 (French, with French summary). MR 1632171, DOI 10.1051/cocv:1998107
J. Ginibre and G. Velo, Sur une équation de Schrödinger non linéaire avec interaction non locale, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. II (Paris, 1979/1980) Res. Notes in Math., vol. 60, Pitman, Boston, Mass.-London, 1982, pp. 155–199, 391–392 (French, with English summary). MR 652511
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582
Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the concentration of blow up solutions for the generalized KdV equation critical in $L^2$, Nonlinear wave equations (Providence, RI, 1998) Contemp. Math., vol. 263, Amer. Math. Soc., Providence, RI, 2000, pp. 131–156. MR 1777639, DOI 10.1090/conm/263/04195
S. Keraani, Études de quelques régimes asymptotiques de l’équation de Schrödinger, Ph.D. thesis, Université Paris-Sud, Orsay, 2000.
Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), no. 2, 353–392. MR 1855973, DOI 10.1006/jdeq.2000.3951
Man Kam Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$, Arch. Rational Mech. Anal. 105 (1989), no. 3, 243–266. MR 969899, DOI 10.1007/BF00251502
F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), no. 2, 427–454. MR 1203233, DOI 10.1215/S0012-7094-93-06919-0
Frank Merle and Pierre Raphael, On universality of blow-up profile for $L^2$ critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), no. 3, 565–672. MR 2061329, DOI 10.1007/s00222-003-0346-z
F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices 8 (1998), 399–425. MR 1628235, DOI 10.1155/S1073792898000270
G. Métivier and S. Schochet, Trilinear resonant interactions of semilinear hyperbolic waves, Duke Math. J. 95 (1998), no. 2, 241–304. MR 1652009, DOI 10.1215/S0012-7094-98-09508-4
A. Moyua, A. Vargas, and L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in $\mathbf R^3$, Duke Math. J. 96 (1999), no. 3, 547–574. MR 1671214, DOI 10.1215/S0012-7094-99-09617-5
U. Niederer, The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials, Helv. Phys. Acta 47 (1974), 167–172. MR 366263
Galina Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré 2 (2001), no. 4, 605–673. MR 1852922, DOI 10.1007/PL00001048
Jeffrey Rauch, Partial differential equations, Graduate Texts in Mathematics, vol. 128, Springer-Verlag, New York, 1991. MR 1223093, DOI 10.1007/978-1-4612-0953-9
Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
T. Tao, Recent progress on the restriction conjecture, arXiv:math.CA/0311181, 2003, Lecture notes, Park City, Utah.
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. MR 2033842, DOI 10.1007/s00039-003-0449-0
Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
Michael I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), no. 5, 545–565. MR 829596, DOI 10.1080/03605308608820435
Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945