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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Book Review

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Book Information:

Author: J. Bourgain
Title: Global solutions of nonlinear Schrödinger equations
Additional book information: Amer. Math. Soc., Providence, RI, 1999, viii+182 pp., ISBN 0-8218-1919-4, $35.00$

References [Enhancements On Off] (What's this?)

1.
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I, Geom. Func. Anal. 3 (1993), 107-156. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II, Geom. Func. Anal. 3 (1993), 209-262. MR 1209299, MR 95d:35160b
2.
J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, IMRN 6 (1996), 277-304. MR 1386079
3.
J. Bourgain, On the growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in ${\mathbb R}^{d}$, J. Analyse Math. 72 (1997), 299-310. MR 1482999
4.
J. Bourgain, On the growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Analyse Math. 77 (1999), 315-348. MR 1753490
5.
J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), no. 2, 256-282. MR 1692486
6.
T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^{s}$, Nonlin. Anal. 14 (1990), 807-836. MR 1055532
7.
T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H\sp 1$, Manuscripta Math. 61, no. 4 (1988), 477-494. MR 0952091
8.
H. Chihara, Local existence for semilinear Schrödinger equations, Math. Japonica 42 (1995), 35-52. MR 1344627
9.
H. Chihara, Smoothing effects of dispersive pseudodifferential equations, preprint (2002).
10.
J. Colliander, J. Delort, C. Kenig, G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), no. 8, 3307-3325. MR 1828607
11.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness for Schrödinger equations with derivatives, SIAM J. Math. Anal. 3 (2001), 649-669.
12.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, A refined global well-posedness result for Schrödinger equations with derivatives, To appear in SIAM J. Math. Anal. (2002).
13.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on ${\mathbb R}$ and ${\mathbb T}$, preprint (2002).
14.
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Almost Conservation Laws and Global Rough Solutions to a Nonlinear Schrödinger Equation, preprint (2002).
15.
P. Constantin, J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), 413-446. MR 0928265
16.
J. Ginibre, An introduction to nonlinear Schrödinger equations, Nonlinear waves (Sapporo, 1995), 85-133. Gakkotosho, Tokyo, 1997. MR 1602772
17.
R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797. MR 0460850
18.
M. Grillakis, Regularity and asymptotic behavior of the wave equation with a critical nonlinearity, Ann. of Math. 132(2) (1990), no. 3, 485-509. MR 1078267
19.
M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307-347. MR 0809718
20.
N.J. Hitchin, G.B. Segal, R.S. Ward, Integrable Systems, Oxford Graduate Texts in Mathematics, 4 (1999). MR 1723384
21.
T. Kato, On the Cauchy problem for (generalized) KdV equation, Advances in Math. Supp. Studies, Studies in Applied Math. 8 (1983), 93-128. MR 0759907
22.
N. Katz, T. Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Lett. 6 (1999), no. 5-6, 625-630. MR 1739220
23.
M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120(5) (1998), 955-980. MR 1646048
24.
C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), 33-69. MR 1101221
25.
C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 1211741
26.
C.E. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 10 (1993), 255-288. MR 1230709
27.
C.E. Kenig, G. Ponce, L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998), 489-545. MR 1660933
28.
C.E. Kenig, G. Ponce, L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106, no. 3 (2001), 617-633. MR 1813239
29.
S. Kuksin, Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs, Comm. Math. Phys. 167 (1995), no. 3, 531-552. MR 1316759
30.
S. Kuksin, On squeezing and flow of energy for nonlinear wave equations, Geom. Funct. Anal. 5 (1995), no. 4, 668-7 MR 1345018.
31.
B. LeMesurier, G. Papanicolaou, C. Sulem, P.-L. Sulem, The focusing singularity of the nonlinear Schrödinger equation, Directions in partial differential equations (Madison, WI, 1985), 159-201, Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, (1987). MR 1013838
32.
H. Lindblad, Sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), no. 2, 503-539. MR 1248683
33.
Y. Martel, F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. 79(9) (2000), no. 4, 339-425. MR 1753061
34.
Y. Martel, F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 (2001), no. 3, 219-254. CMP 2002:11
35.
Y. Martel, F. Merle, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), no. 1, 74-123. MR 1829643
36.
F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), 223-240. MR 1048692
37.
F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), 427-453. MR 1203233
38.
F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), no. 3, 555-578. MR 1824989
39.
F. Merle, P. Raphael, Sharp upper bound on the blow-up rate for critical nonlinear Schrödinger equation, preprint (2002).
40.
P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715. MR 0904948
41.
G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J. 86 (1997), 109-142. MR 1427847
42.
G. Staffilani, Quadratic forms for a 2D Schrödinger equation, Duke Math. J. 86 (1997), 79-107. MR 1427846
43.
E. Stein, Harmonic Analysis, Princeton Mathematical Series 43 (1993). MR 1232192
44.
W. Strauss, Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, 73 (1989). MR 1032250
45.
C. Sulem, P.-L. Sulem, The nonlinear Schrödinger equation, Applied Math Scien. 139, Springer, (1999). MR 1696311
46.
Y. Tsutsumi, $L\sp 2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30, no. 1, (1987), 115-125. MR 0915266
47.
L. Vega, Doctoral Thesis, Universidad Autonima de Madrid, Spain (1987).
48.
L. Vega, The Schrödinger equation: pointwise convergence to the initial data, Proc. AMS. 102 (1988), 874-878. MR 0934859
49.
T. Wolff, A Kakeya-type problem for circles, Amer. J. Math. 119 (1997), no. 5, 985-1026. MR 1473067
50.
T. Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996), 129-162, Amer. Math. Soc., Providence, RI, 1999. MR 1660476
51.
V.E. Zakharov, Collapse of Langmuir waves, Sov. Phys. JETP 35 (1997), 908-914.
52.
V.E. Zakharov, V.S. L'vov, G. Falkovich, Kolmogorov spectra of turbulence, Springer series in nonlinear dynamics, Berlin ; New York : Springer-Verlag, (1992).
Review Information:

Reviewer: Gigliola Staffilani
Affiliation: Massachusetts Institute of Technology
Email: gigliola@math.mit.edu
Journal: Bull. Amer. Math. Soc. 40 (2003), 99-107
Published electronically: October 16, 2002
Review copyright: © Copyright 2002 American Mathematical Society