Bilinear estimates and applications to 2d NLS
Authors:
J. E. Colliander, J.M. Delort, C. E. Kenig and G. Staffilani
Journal:
Trans. Amer. Math. Soc. 353 (2001), 33073325
MSC (2000):
Primary 35Q55, 42B35
Published electronically:
April 10, 2001
MathSciNet review:
1828607
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The three bilinearities for functions are sharply estimated in function spaces associated to the Schrödinger operator . These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy globalintime and blowup solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.
 1.
J.
Bourgain, Fourier transform restriction phenomena for certain
lattice subsets and applications to nonlinear evolution equations. I.
Schrödinger equations, Geom. Funct. Anal. 3
(1993), no. 2, 107–156. MR 1209299
(95d:35160a), http://dx.doi.org/10.1007/BF01896020
J.
Bourgain, Fourier transform restriction phenomena for certain
lattice subsets and applications to nonlinear evolution equations. II. The
KdVequation, Geom. Funct. Anal. 3 (1993),
no. 3, 209–262. MR 1215780
(95d:35160b), http://dx.doi.org/10.1007/BF01895688
 2.
Jean
Bourgain, On the growth in time of higher Sobolev norms of smooth
solutions of Hamiltonian PDE, Internat. Math. Res. Notices
6 (1996), 277–304. MR 1386079
(97k:35016), http://dx.doi.org/10.1155/S1073792896000207
 3.
J.
Bourgain, Refinements of Strichartz’ inequality and
applications to 2DNLS with critical nonlinearity, Internat. Math.
Res. Notices 5 (1998), 253–283. MR 1616917
(99f:35184), http://dx.doi.org/10.1155/S1073792898000191
 4.
J.
Bourgain, Global solutions of nonlinear Schrödinger
equations, American Mathematical Society Colloquium Publications,
vol. 46, American Mathematical Society, Providence, RI, 1999. MR 1691575
(2000h:35147)
 5.
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
(91j:35252), http://dx.doi.org/10.1016/0362546X(90)90023A
 6.
J. E. Colliander, C. E. Kenig, and G. Staffilani, An Space Approach to Local Wellposedness of the KPI Equation, in preparation.
 7.
J.M. Delort and D. Fang, Almost global existence for solutions of semilinear KleinGordon equations with small weakly decaying Cauchy data 25 (2000), no. 1112, 21192169.
 8.
Carlos
E. Kenig, Gustavo
Ponce, and Luis
Vega, A bilinear estimate with applications
to the KdV equation, J. Amer. Math. Soc.
9 (1996), no. 2,
573–603. MR 1329387
(96k:35159), http://dx.doi.org/10.1090/S0894034796002007
 9.
Carlos
E. Kenig, Gustavo
Ponce, and Luis
Vega, Quadratic forms for the 1D semilinear
Schrödinger equation, Trans. Amer. Math.
Soc. 348 (1996), no. 8, 3323–3353. MR 1357398
(96j:35233), http://dx.doi.org/10.1090/S0002994796016455
 10.
B.
LeMesurier, G.
Papanicolaou, C.
Sulem, and P.L.
Sulem, The focusing singularity of the nonlinear Schrödinger
equation, Directions in partial differential equations (Madison, WI,
1985) Publ. Math. Res. Center Univ. Wisconsin, vol. 54, Academic
Press, Boston, MA, 1987, pp. 159–201. MR 1013838
(90i:35249)
 11.
Kenji
Nakanishi, Energy scattering for nonlinear KleinGordon and
Schrödinger equations in spatial dimensions 1 and 2, J. Funct.
Anal. 169 (1999), no. 1, 201–225. MR 1726753
(2000m:35141), http://dx.doi.org/10.1006/jfan.1999.3503
 12.
Gigliola
Staffilani, On the growth of high Sobolev norms of solutions for
KdV and Schrödinger equations, Duke Math. J. 86
(1997), no. 1, 109–142. MR 1427847
(98b:35192), http://dx.doi.org/10.1215/S001270949708604X
 13.
Gigliola
Staffilani, Quadratic forms for a 2D semilinear Schrödinger
equation, Duke Math. J. 86 (1997), no. 1,
79–107. MR
1427846 (98b:35191), http://dx.doi.org/10.1215/S0012709497086038
 14.
T. Tao, Multilinear weighted convolution of functions and applications to nonlinear dispersive equations, to appear, Amer. J. Math.
 1.
 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I,II, Geom. Funct. Anal. 3 (1993), 107156, 209262. MR 95d:35160a MR 95d:35160b
 2.
 , On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 6 (1996), 277304. MR 97k:35016
 3.
 , Refinements of Strichartz' inequality and applications to 2DNLS with critical nonlinearity, International Mathematical Research Notices 5 (1998), 253283. MR 99f:35184
 4.
 , Global solutions of nonlinear Schrödinger equations, American Mathematical Society, Colloquium Publications, 46, Providence, RI, 1999. MR 2000h:35147
 5.
 T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in , Nonlinear Anal. 14 (1990), 807836. MR 91j:35252
 6.
 J. E. Colliander, C. E. Kenig, and G. Staffilani, An Space Approach to Local Wellposedness of the KPI Equation, in preparation.
 7.
 J.M. Delort and D. Fang, Almost global existence for solutions of semilinear KleinGordon equations with small weakly decaying Cauchy data 25 (2000), no. 1112, 21192169.
 8.
 C. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573603. MR 96k:35159
 9.
 , Quadratic forms for the 1D semilinear Schrödinger equation., Trans. Amer. Math. Soc. 348 (1996), 33233353. MR 96j:35233
 10.
 B. LeMesurier, G. Papanicolaou, C. Sulem, and P.L. Sulem, The focusing singularity of the nonlinear Schrödinger equation, Directions in partial differential equations (Madison, WI, 1985), Academic Press, Boston, MA, 1987, pp. 159201. MR 90i:35249
 11.
 K. Nakanishi, Energy scattering for nonlinear KleinGordon and Schrödinger equations in spatial dimensions and , J. Funct. Anal. 169 (1999), 201225. MR 2000m:35141
 12.
 G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J. 86 (1997), 109142. MR 98b:35192
 13.
 , Quadratic forms for a D semilinear Schrödinger equation, Duke Math. J. 86 (1997), 79107. MR 98b:35191
 14.
 T. Tao, Multilinear weighted convolution of functions and applications to nonlinear dispersive equations, to appear, Amer. J. Math.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35Q55,
42B35
Retrieve articles in all journals
with MSC (2000):
35Q55,
42B35
Additional Information
J. E. Colliander
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Email:
colliand@math.berkeley.edu
J.M. Delort
Affiliation:
Département of Mathématiques, Université de ParisNord, 93430 Villetaneuse, France
Email:
delort@math.univparis13.fr
C. E. Kenig
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email:
cek@math.uchicago.edu
G. Staffilani
Affiliation:
Department of Mathematics, Stanford University, Stanford California 94305
Email:
gigliola@math.stanford.edu
DOI:
http://dx.doi.org/10.1090/S000299470102760X
PII:
S 00029947(01)02760X
Keywords:
Nonlinear Schr\"odinger equation,
nonlinear dispersive equations,
weak turbulence,
NLS blowup,
bilinear estimates,
multilinear harmonic analysis,
Strichartz inequalities
Received by editor(s):
July 24, 2000
Published electronically:
April 10, 2001
Additional Notes:
J.E.C. was supported in part by an N.S.F. Postdoctoral Research Fellowship.
C.E.K. was supported in part by N.S.F. Grant DMS 9500725
G.S. was supported in part by N.S.F. Grant DMS 9800879
Article copyright:
© Copyright 2001
American Mathematical Society
