American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.   Google Scholar

[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. Google Scholar

[13]

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

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[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.   Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

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

[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.   Google Scholar

[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. Google Scholar

[13]

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

[1]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905
[2]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
[3]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
[4]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
[5]

Dan-Andrei Geba, Evan Witz. Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations. Electronic Research Archive, 2020, 28 (2) : 627-649. doi: 10.3934/era.2020033
[6]

Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177
[7]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
[8]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[9]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[10]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[12]

Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061
[13]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[14]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[15]

Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699
[16]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[17]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
[18]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
[19]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016
[20]

Claudianor O. Alves, Chao Ji. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2671-2685. doi: 10.3934/dcds.2020145

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (139)
  • HTML views (166)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy