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Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity
A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates
182 Memorial Dr, Cambridge, MA 02142, USA |
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.
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] | |
[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] | |
[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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