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March  2018, 17(2): 627-646. doi: 10.3934/cpaa.2018034

A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates

Felipe Hernandez ,

182 Memorial Dr, Cambridge, MA 02142, USA

* Corresponding author:Felipe Hernandez

Received  December 2015 Revised  September 2017 Published  March 2018

A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new proof of the bilinear Strichartz estimate as well as the multilinear restriction theorem for the paraboloid.

Keywords: Schrödinger equation, bilinear Strichartz, multilinear restriction.
Mathematics Subject Classification: Primary:35Q41;Secondary:42B37.
Citation: Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure and Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034
References:
[1]

J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and {K}akeya conjectures, Acta Mathematica, 196 (2006), 261-302. 

[2]

J. Bourgain and C. Demeter, The proof of the $\ell^2$ decoupling conjecture, arXiv preprint arXiv: 1403.5335, 2014.

[3]

J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geometric and Functional Analysis, 21 (2011), 1239-1295. 

[4]

J. Bourgain, Refinements of {S}trichartz' inequality and applications to 2D-NLS with critical nonlinearity, Intern. Mat. Res. Notices, 5 (1998), 253-283. 

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservations laws and global rough sol.utions to a nonlinear Schrödinger equation, Math. Res. Letters, 9 (2002), 659-682. 

[6]

L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics, 8 (1956), 399-404. 

[7]

L. Guth, A short proof of the multilinear Kakeya inequality, In Mathematical Proceedings of the Cambridge Philosophical Society, volume 158, pages 147-153. Cambridge Univ Press, 2015.

[8]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis and PDE, 5 (2012), 339-362. 

[9]

Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on closed manifolds, Communications in Partial Differential Equations, 37 (2012), 1186-1236. 

[10]

S. Joseph, The max-flow min-cut theorem, 2007.

[11]

S. Klainerman, I. Rodnianski and T. Tao, A physical space approach to wave equation bilinear estimates, Journal d'Analyse Mathématique, 87 (2002), 299-336. 

[12]

A. Staples-Moore, Network flows and the max-flow min-cut theorem, http://www.math.uchicago.edu/may/VIGRE/VIGRE2009/REUPapers/Staples-Moore.pdf.

[13]

T. Tao, A physical space proof of the bilinear Strichartz and local smoothing estimate for the Schrödinger equation, 2010.

show all references

References:
[1]

J. Bennett, A. Carbery and T. Tao, On the multilinear restriction and {K}akeya conjectures, Acta Mathematica, 196 (2006), 261-302. 

[2]

J. Bourgain and C. Demeter, The proof of the $\ell^2$ decoupling conjecture, arXiv preprint arXiv: 1403.5335, 2014.

[3]

J. Bourgain and L. Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geometric and Functional Analysis, 21 (2011), 1239-1295. 

[4]

J. Bourgain, Refinements of {S}trichartz' inequality and applications to 2D-NLS with critical nonlinearity, Intern. Mat. Res. Notices, 5 (1998), 253-283. 

[5]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservations laws and global rough sol.utions to a nonlinear Schrödinger equation, Math. Res. Letters, 9 (2002), 659-682. 

[6]

L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics, 8 (1956), 399-404. 

[7]

L. Guth, A short proof of the multilinear Kakeya inequality, In Mathematical Proceedings of the Cambridge Philosophical Society, volume 158, pages 147-153. Cambridge Univ Press, 2015.

[8]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Analysis and PDE, 5 (2012), 339-362. 

[9]

Z. Hani, Global well-posedness of the cubic nonlinear Schrödinger equation on closed manifolds, Communications in Partial Differential Equations, 37 (2012), 1186-1236. 

[10]

S. Joseph, The max-flow min-cut theorem, 2007.

[11]

S. Klainerman, I. Rodnianski and T. Tao, A physical space approach to wave equation bilinear estimates, Journal d'Analyse Mathématique, 87 (2002), 299-336. 

[12]

A. Staples-Moore, Network flows and the max-flow min-cut theorem, http://www.math.uchicago.edu/may/VIGRE/VIGRE2009/REUPapers/Staples-Moore.pdf.

[13]

T. Tao, A physical space proof of the bilinear Strichartz and local smoothing estimate for the Schrödinger equation, 2010.

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