Global solutions of nonlinear Schrödinger equations

193 Pages·1999·13.679 MB·English

by  Jean Bourgain

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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
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44 Max-Alber t Knus , Alexander Merkurjev, an d Markus Rost, Th e book of
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43 Lui s A. Caffarell i an d Xavier Cabre, Full y nonlinear elliptic equations, 1995
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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
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17 J . M. Wedderburn, Lecture s on matrices, 1934
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9 G . D. Birkhoff, Dynamica l systems, 1927
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6 G . C . Evans, Th e logarithmic potential, discontinuous Dirichlet and Neumann problems,
1927
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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
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Republication, systematic copying, or multiple reproduction of any material in this publication
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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