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), 3307-3325

MSC (2000):
Primary 35Q55, 42B35

Published electronically:
April 10, 2001

MathSciNet review:
1828607

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 global-in-time and blow-up 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**, 10.1007/BF01896020

J. Bourgain,*Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation*, Geom. Funct. Anal.**3**(1993), no. 3, 209–262. MR**1215780**, 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**, 10.1155/S1073792896000207**3.**J. Bourgain,*Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity*, Internat. Math. Res. Notices**5**(1998), 253–283. MR**1616917**, 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****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**, 10.1016/0362-546X(90)90023-A**6.**J. E. Colliander, C. E. Kenig, and G. Staffilani,*An Space Approach to Local Wellposedness of the KP-I Equation*, in preparation.**7.**J.-M. Delort and D. Fang,*Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data***25**(2000), no. 11-12, 2119-2169.**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**, 10.1090/S0894-0347-96-00200-7**9.**Carlos E. Kenig, Gustavo Ponce, and Luis Vega,*Quadratic forms for the 1-D semilinear Schrödinger equation*, Trans. Amer. Math. Soc.**348**(1996), no. 8, 3323–3353. MR**1357398**, 10.1090/S0002-9947-96-01645-5**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****11.**Kenji Nakanishi,*Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2*, J. Funct. Anal.**169**(1999), no. 1, 201–225. MR**1726753**, 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**, 10.1215/S0012-7094-97-08604-X**13.**Gigliola Staffilani,*Quadratic forms for a 2-D semilinear Schrödinger equation*, Duke Math. J.**86**(1997), no. 1, 79–107. MR**1427846**, 10.1215/S0012-7094-97-08603-8**14.**T. Tao,*Multilinear weighted convolution of functions and applications to nonlinear dispersive equations*, to appear, Amer. J. Math.

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 Paris-Nord, 93430 Villetaneuse, France

Email:
delort@math.univ-paris13.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:
https://doi.org/10.1090/S0002-9947-01-02760-X

Keywords:
Nonlinear Schr\"odinger equation,
nonlinear dispersive equations,
weak turbulence,
NLS blow-up,
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