Table Of ContentSelected Title s i n Thi s Serie s
46 J . Bourgain, Globa l solutions of nonlinear Schrodinger equations, 1999
45 Nichola s M. Katz and Peter Sarnak, Rando m matrices, Frobenius eigenvalues, and
monodromy, 199 9
44 Max-Alber t Knus , Alexander Merkurjev, an d Markus Rost, Th e book of
involutions, 1998
43 Lui s A. Caffarell i an d Xavier Cabre, Full y nonlinear elliptic equations, 1995
42 Victo r Guillemin and Shlomo Sternberg, Variation s on a theme by Kepler, 1990
41 Alfre d Tarski and Steven Givant, A formalization of set theory without variables, 1987
40 R . H. Bing, Th e geometric topology of 3-manifolds, 198 3
39 N . Jacobson, Structur e and representations of Jordan algebras, 1968
38 O . Ore, Theor y of graphs, 1962
37 N . Jacobson, Structur e of rings, 1956
36 W . H . Gottschalk an d G. A. Hedlund, Topologica l dynamics, 1955
35 A . C . Schaeffer an d D. C. Spencer, Coefficien t region s for Schlicht functions, 195 0
34 J . L. Walsh, Th e location of critical points of analytic and harmonic functions, 195 0
33 J . F. Ritt, Differentia l algebra, 1950
32 R . L. Wilder, Topolog y of manifolds, 1949
31 E . Hille and R. S . Phillips, Functiona l analysis and semigroups, 1957
30 T . Rado, Lengt h and area, 1948
29 A . Weil, Foundation s of algebraic geometry, 1946
28 G . T. Whyburn, Analyti c topology, 1942
27 S . Lefschetz, Algebrai c topology, 1942
26 N . Levinson, Ga p and density theorems, 1940
25 Garret t Birkhoff , Lattic e theory, 1940
24 A . A. Albert, Structur e of algebras, 1939
23 G . Szego , Orthogona l polynomials, 1939
22 C . N. Moore, Summabl e series and convergence factors, 1938
21 J . M. Thomas, Differentia l systems, 1937
20 J . L. Walsh, Interpolatio n and approximation by rational functions in the complex
domain, 193 5
19 R . E. A. C. Paley and N. Wiener, Fourie r transforms in the complex domain, 1934
18 M . Morse, Th e calculus of variations in the large, 1934
17 J . M. Wedderburn, Lecture s on matrices, 1934
16 G . A. Bliss, Algebrai c functions, 193 3
15 M . H. Stone, Linea r transformations in Hilbert space and their applications to analysis,
1932
14 J . F. Ritt, Differentia l equations from the algebraic standpoint, 193 2
13 R . L . Moore, Foundation s of point set theory, 1932
12 S . Lefschetz, Topology , 1930
11 D . Jackson, Th e theory of approximation, 193 0
10 A . B. Coble, Algebrai c geometry and theta functions, 192 9
9 G . D. Birkhoff, Dynamica l systems, 1927
8 L . P. Eisenhart, Non-Riemannia n geometry, 1927
7 E . T. Bell, Algebrai c arithmetic, 1927
6 G . C . Evans, Th e logarithmic potential, discontinuous Dirichlet and Neumann problems,
1927
(Continued in the back of this publication)
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Globa l Solution s of
Nonlinea r Schrodinge r
Equation s
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http://dx.doi.org/10.1090/coll/046
America n Mathematica l Societ y
Colloquiu m Publication s
Volum e 46
Global Solutions of
Nonlinear Schrodinger
Equations
J. Bourgai n
America n Mathematica l Societ y
Providence , Rhod e Island
Editorial Boar d
Joan S. Birman
Susan J. Friedlander, Chair
Stephen Lichtenbau m
1991 Mathematics Subject Classification. Primar y 35Q55 .
ABSTRACT. Th e aim of this book is to describe recent progress on various issues in the theor y
of nonlinear dispersiv e equations, primarily the nonlinear Schrodinge r equatio n (NLS) . In par -
ticular, th e Cauchy problem for the defocusing critical NLS with radial data is discussed. Ne w
techniques and results are described on global existence of large data solutions below the energy
norm. Curren t research in Harmonic Analysis around Strichartz' inequalities and its relevance to
nonlinear PDE is presented. Also several topics in NLS theory on bounded domains are reviewed.
In this respect, a partial survey is given of the theory of invariant Gibb s measures and recen t
developments in KAM theory for PDE's.
Library o f Congress Cataloging-in-Publicatio n Dat a
Bourgain, Jean, 1954-
Global solutions of nonlinear Schrodinger equations / Jea n Bourgain.
p. cm. — (American Mathematical Society colloquium publications, ISSN 0065-9258 ; v. 46)
Includes bibliographical references and index.
ISBN 0-8218-1919- 4
1. Schrodinger equation. 2 . Differential equations, Partial—Numerical solutions. 3 . Nonlin-
ear theories. I . Title. II . Series: Colloquiu m publications (America n Mathematical Society) ;
v. 46.
QC174.26.W28B68 199 9
bib'.353—dc21 99-1306 6
CIP
Copying an d reprinting . Individua l reader s of this publication, an d nonprofi t librarie s
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permissio n is granted to quote brief passages from this publication i n
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society .
Requests for such permission should be addressed to the Assistant to the Publisher, America n
Mathematical Society , P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also
be made by e-mail to [email protected] .
© 199 9 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government .
Printed in the United States of America.
@ Th e paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
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10 9 8 7 6 5 4 3 2 1 0 4 03 02 01 00 99
Contents
Introduction and summary 1
I. An overview of results on the Cauchy problem for NLS 5
1. Equations 5
2. Wellposedness of the Cauchy problem 6
3. Scatterin g results 1 0
4. Estimate s on the linear group 1 1
5. Solvin g the Cauchy problem 1 4
6. Derivative nonlinear Schrodinger equations 1 4
II. Further comments 1 7
1. Construction of blowup solutions for conformal NLS from the ground
state 1 7
2. Behaviour of higher Sobolev norms 2 8
3. Fourie r restriction theory beyond 1? 3 3
4. L 2-concentration phenomenon 3 7
5. The Schrodinger maximal function 4 5
6. Derivative nonlinear Schrodinger equation 4 8
III. 3D ./^-critical defocusing NLS 5 1
Consider 3D NLS 5 1
Fix a time interval I = [0, T) 5 1
Sketch of the argument 5 4
A concentration property 5 9
A version of Morawetz inequality 6 1
Construction of an appropriate time interval 6 3
Details on the perturbative analysis 66
A variant of the method 7 0
IV. Global wellposedness below energy norm 7 9
1. Description of the method 7 9
2. The example of the NLW 7 9
3. Th e case of the nonlinear Schrodinger equation 8 3
4. Symplecti c capacities and symplectic Hilbert spaces 9 3
5. Globa l wellposedness of the NLW (4.41) 9 7
V. Nonlinear Schrodinger equation with periodic boundary conditions 10 5
1. Introduction 10 5
2. Results on the Cauchy problem 10 6
3. Periodi c Strichartz inequalities 10 8
viii CONTENT S
4. Sketc h of proof of Theorems 2.1 and 2.7
5. Invariant Gibbs measures (ID )
6. Invarinat Gibbs measures (D > 1)
7. Invariant Gibbs measures (unbounded domains)
8. Quasi-periodi c solutions
Appendix 1. Growth of Sobolev norms in linear Schrodinger equations
smooth time dependent potentia l
Appendix 2. Zakharov systems
References
Index
http://dx.doi.org/10.1090/coll/046/01
0. Introductio n and summar y
Despite the attention this theory has received over recent years, there are many
problems left essentiall y unsolved concerning the longtime behaviour of solutions
to the Cauchy problem for the nonlinear Schrodinger equation (NLS for short)
iu + Au±u\u\p'2 = 0
t
u{0) = (/)eH s{Rd) (o-i)
with Hamiltonian
-I
H(4>) >I2^I dx. (0.2 )
Although the initial value problem (IVP ) theory is satisfactory fo r local time be-
haviour and small data, many issues on the behaviour of solutions for large dat a
are far from understood. I n the following case (i.e . "+ " sig n in (0.1)), it is well
known that fo r p > 2 + |, smoot h solutions of (0.1) may blowup in finite time .
There is a vast problematic in this context, concerning questions such as blowup
speed, blowu p profile and its stability etc. , pursued bot h purely mathematicall y
and numerically. Thi s body of problems will not be our primarily issue here and we
will only comment on a few aspects of recent research. We will rather concentrat e
on equation (0,1 ) in the defocusing case, when it is expected that local solutions
extend to global ones and preserve their i/s-class for all time, with scattering be-
haviour for sufficiently hig h degree nonlinearity. W e are particularly interested in
two problems that we describe briefly next.
(i) The /^-critical equatio n
Consider the NLS
iu + Au - u\u\ p-2 =0 , p = 2+—^- (d>3 ) (0.3 )
t
a — 2
for which the homogeneous H1 -space H1 i s the scale invariant Sobolev space. It is
known that there is local wellposedness for any data< /> G Hs, s > 1 and the result
is global for data small in H1. I t is an open problem whether classical solution s
exist global in time. Remark that since the Hamiltonian (and the L2-norm) provide
the only apriori bounds on the solution, also a classical theory needs to include a
i
