Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bf8c574d5-956mj Total loading time: 0 Render date: 2025-02-22T21:46:54.333Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  28 March 2019

Benjamin Dodson
Benjamin Dodson
Affiliation:
The Johns Hopkins University
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Defocusing Nonlinear Schrödinger Equations , pp. 236 - 240
Publisher: Cambridge University Press
Print publication year: 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berestycki, H. and Lions, P. L. (1979). Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein–Gordon. C. R. Acad. Sci. Paris Sér. A–B, 288(7), A395–A398.Google Scholar
Bergh, J. and Löfstrom, J. (1976). Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Berlin-New York: Springer.Google Scholar
Bourgain, J. (1998). Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Internat. Math. Res. Notices, 1998(5), 253–283.Google Scholar
Bourgain, J. (1999). Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Amer. Math. Soc., 12(1), 145–171.CrossRefGoogle Scholar
Cazenave, T. and Weissler, F. B. (1988). The Cauchy problem for the nonlinear Schrödinger equation in H1. F.B. Manuscripta Math., 61, 477–494.Google Scholar
Christ, M., Colliander, J., and Tao, T. (2003). Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math., 125(6), 1235–1293.Google Scholar
Christ, M., Colliander, J., and Tao, T. (2008). A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order. J. Funct. Anal., 254(2), 368–395.CrossRefGoogle Scholar
Christ, M. and Kiselev, A. (2001). Maximal functions associated to filtrations. J. Funct. Anal., 179(2), 409–425.Google Scholar
Colliander, J., Grillakis, M., and Tzirakis, N. (2007). Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on R2. Internat. Math. Res. Notices, 23(1), 90–119.Google Scholar
Colliander, J., Grillakis, M., and Tzirakis, N. (2009). Tensor products and correlation estimates with applications to nonlinear Schrödinger equations. Comm. Pure Appl. Math., 62(7), 920–968.Google Scholar
Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T. (2004). Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3. Comm. Pure Appl. Math., 21, 987–1014.Google Scholar
Colliander, J., Keel, M., Staffilani, G., Takaoka, H., and Tao, T. (2008). Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation on R3. Ann. of Math. (2), 167, 767–865.Google Scholar
Colliander, J. and Roy, T. (2011). Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on R2. Commun. Pure Appl. Anal., 10(2), 397–414.Google Scholar
Conway, J. B. (1990). A Course in Functional Analysis, 2nd edn, Graduate Texts in Mathematics, vol. 96, New York: Springer.Google Scholar
Dodson, B. (2011). Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Commun. Pure Appl. Anal., 10(1), 127–140.Google Scholar
Dodson, B. (2012). Global well-posedness and scattering for the defocusing L2-critical nonlinear Schrödinger equation when d ≥ 3. J. Amer. Math. Soc., 25(2), 429–463.Google Scholar
Dodson, B. (2015). Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state. Adv. Math., 285, 1589–1618.Google Scholar
Dodson, B. (2016a). Global well-posedness and scattering for the defocusing L2-critical nonlinear Schrödinger equation when d = 1. Amer. J. Math., 138(2), 531–569.Google Scholar
Dodson, B. (2016b). Global well-posedness and scattering for the defocusing L2-critical nonlinear Schrödinger equation when d = 2. Duke Math. J., 165(18), 3435–3516.Google Scholar
Duyckaerts, T., Holmer, J., and Roudenko, S. (2008). Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett., 15(5–6), 1233–1250.Google Scholar
Ginibre, J. and Velo, G. (1992). Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys., 144(1), 163–188.Google Scholar
Grafakos, L. (2004). Classical Fourier Analysis, 3rd edn, Graduate Texts in Mathematics, vol. 249, New York: Springer.Google Scholar
Grillakis, M. (2000). On nonlinear Schrödinger equations. Comm. Partial Differential Equations 25(9–10), 1827–1844.CrossRefGoogle Scholar
Guevara, C. (2014). Global behavior of finite energy solutions to the focusing nonlinear Schrödinger equation in d dimensions. Appl. Math. Res. Express, 2014, 177–243.Google Scholar
Hadac, M., Herr, S., and Koch, H. (2009). Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(3), 917–941.Google Scholar
Holmer, J., Platte, R., and Roudenko, S. (2010). Blow-up criteria for the 3D cubic nonlinear Schrödinger equation. Nonlinearity, 23(4), 977–1030.Google Scholar
Holmer, J. and Roudenko, S. (2008). A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm. Math. Phys., 282(2), 435–467.Google Scholar
Keel, M. and Tao, T. (1998). Endpoint Strichartz estimates. Amer.J. Math., 120(4–6), 945–957.Google Scholar
Kenig, C. and Merle, F. (2006). Global well-posedness, scattering, and blow-up for the energy-critical, focusing nonlinear Schrödinger equation in the radial case. Invent. Math., 166(3), 645–675.Google Scholar
Kenig, C. and Merle, F. (2010). Scattering for Ḣ1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc., 362(4), 1937–1962.Google Scholar
Keraani, S. (2001). On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations, 175(2), 353–392.Google Scholar
Killip, R., Tao, T., and Visan, M. (2009). The cubic nonlinear Schrödinger equation in two dimensions with radial data. J. Eur. Math. Soc. (JEMS), 11(6), 1203–1258.Google Scholar
Killip, R. and Visan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Amer. J. Math., 132(2), 361–424.Google Scholar
Killip, R. and Visan, M. (2012). Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Anal. PDE, 5(4), 855–885.Google Scholar
Killip, R. and Visan, M. (2013). Nonlinear Schrödinger equations at critical regularity. In Clay Math. Proceedings, vol. 17, Providence, RI: American Mathematical Society, pp. 325–437.Google Scholar
Killip, R., Visan, M., and Zhang, X. (2008). The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE, 1(2), 229– 266.Google Scholar
Koch, H. and Tataru, D. (2007). A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Internat. Math. Res. Notices, 16, Art. ID rnm053, 36 pp.Google Scholar
Koch, H. and Tataru, D. (2012). Energy and local energy bounds for the 1-D cubic NLS equation in H−1/4. Ann. Inst. H. Poincaré Anal. Non Linéaire, 29(6), 955–988.CrossRefGoogle Scholar
Lieb, E. and Loss, M. (2001). Analysis, 2nd edn, Graduate Studies in Mathematics, vol. 14, Providence, RI: American Mathematical Society.Google Scholar
Lin, J. and Strauss, W. (1978). Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal., 30(2), 245–263.CrossRefGoogle Scholar
Montgomery-Smith, J. S., (1998). Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math. J., 91(2), 393–408.Google Scholar
Murphy, J. (2014). Inter-critical NLS: critical Ḣs bounds imply scattering. SIAM J. Math. Anal., 46(1), 939–997.Google Scholar
Murphy, J. (2015). The radial, defocusing, nonlinear Schrödinger equation in three dimensions. Comm. Partial Differential Equations, 40(2), 265–308.Google Scholar
Muscalu, C. and Schlag, W. (2013). Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, vol. 137, Cambridge: Cambridge University Press.Google Scholar
Planchon, F. and Vega, L. (2009). Bilinear virial identities and applications. Ann. Sci. École Norm. Sup. (4), 42(2), 261–290.Google Scholar
Ryckman, E. and Visan, M. (2007). Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R1+4. Amer. J. Math., 129(1), 1–60.Google Scholar
Shao, S. (2009). A note on the restriction conjecture in the cylindrically symmetric case. Proc. Amer. Math. Soc., 137(1), 135–143.Google Scholar
Smith, H. F. and Sogge, C. D. (2000). Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations, 25(11–12), 2171–2183.Google Scholar
Sogge, C. D. (1993). Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge: Cambridge University Press.Google Scholar
Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press.Google Scholar
Stein, E. M. (1993). Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton University Press.Google Scholar
Strichartz, R. S. (1977). Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J., 44(3), 705–714.Google Scholar
Sulem, C. and Sulem, P. L. (1999). The Nonlinear Schrödinger Equation, Applied Mathematical Sciences, vol. 139, New York: Springer.Google Scholar
Tao, T. (2000). Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm. Partial Differential Equations, 25(7– 8), 1471–1485.Google Scholar
Tao, T. (2003). A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal., 13(6), 1359–1384.Google Scholar
Tao, T. (2005). Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data. New York J. Math., 11, 57–80.Google Scholar
Tao, T. (2006). Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, vol. 104, Washington, DC: Conference Board of the Mathematical Sciences.Google Scholar
Tao, T. and Visan, M. (2005). Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differential Equations, 118, 28 pp.Google Scholar
Tao, T., Visan, M., and Zhang, X. (2007a). The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm. Partial Differential Equations, 32(7– 9), 1281–1343.Google Scholar
Tao, T., Visan, M., and Zhang, X. (2007b). Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions. Duke Math. J., 140(1), 165–202.Google Scholar
Tao, T., Visan, M., and Zhang, X. (2008). Minimal-mass blowup solutions of the mass-critical NLS. Forum Math., 20(5), 881–919.Google Scholar
Tataru, D. (1998). Local and global results for wave maps I. Comm. Partial Differential Equations, 23(9–10), 1781–1793.Google Scholar
Taylor, M. E. (2000). Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs, vol. 81, Providence, RI: American Mathematical Society.Google Scholar
Taylor, M. E. (2006). Measure Theory and Integration, Graduate Studies in Mathematics, vol. 6, Providence, RI: American Mathematical Society.Google Scholar
Taylor, M. E. (2011). Partial Differential Equations I–III, 2nd edn, Applied Mathematical Sciences, vol. 115, New York: Springer.Google Scholar
Visan, M. (2006). The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Ph.D. thesis, UCLA.Google Scholar
Visan, M. (2007). The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J., 138, 281–374.Google Scholar
Visan, M. (2012). Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. Internat. Math. Res. Notices 2, 5, 1037–1067.Google Scholar
Weinstein, M. (1982). Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys., 87(4), 567–576.Google Scholar
Weinstein, M. (1989). The nonlinear Schrödinger equation–singularity formation, stability and dispersion. In The Connection between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemporary Mathematics, vol. 99, Providence, RI: American Mathematical Society, pp. 213– 232.Google Scholar
Yajima, K. (1987). Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys., 110(3), 415–426.Google Scholar
Yosida, K. (1980). Functional Analysis, 6th edn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Berlin: Springer.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Benjamin Dodson, The Johns Hopkins University
  • Book: Defocusing Nonlinear Schrödinger Equations
  • Online publication: 28 March 2019
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Benjamin Dodson, The Johns Hopkins University
  • Book: Defocusing Nonlinear Schrödinger Equations
  • Online publication: 28 March 2019
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Benjamin Dodson, The Johns Hopkins University
  • Book: Defocusing Nonlinear Schrödinger Equations
  • Online publication: 28 March 2019
Available formats
×