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Articles

Almost sure scattering for the energy-critical NLS with radial data below H1(R4)

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Pages 51-71 | Received 29 Apr 2018, Accepted 21 Sep 2018, Published online: 15 Feb 2019

References

  • Ryckman, E., 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.
  • Visan, M. (2012). Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. Int. Math. Res. Not. (5):1037–1067.
  • Christ, M., Colliander, J., Tao, T. (2003). Ill-posedness for nonlinear Schrödinger and wave equations. Preprint arXiv:math/0311048.
  • Nahmod, A., Staffilani, G. (2015). Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space. J. Eur. Math. Soc. 17(7):1687–1759.
  • Pocovnicu, O. (2017). Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on Rd, d = 4 and 5. J. Eur. Math. Soc. 19:2521–2575.
  • Oh, T., Pocovnicu, O. (2016). Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R3. J. Math. Pures Appl. 105(3):342–366.
  • Benyi, A., Oh, T., Pocovnicu, O. (2015). On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on Rd,d≥3. Trans. Amer. Math. Soc. Ser. B. 2:1–50.
  • Brereton, J. T. (2019). Almost Sure Local Well-Posedness for the Supercritical Quintic NLS. Tunis. J. Math. 1(3), 427–453.
  • Dodson, B., Lührmann, J., Mendelson, D. (2017). Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data. Amer. J. Math. Preprint arXiv:1703.09655.
  • Burq, N., Thomann, L., Tzvetkov, N. (2013). Long time dynamics for the one dimensional nonlinear Schrödinger equation. Ann. Inst. Fourier (Grenoble). 63(6):2137–2198.
  • Deng, Y. (2012). Two-dimensional nonlinear Schrödinger equation with random radial data. Anal. PDE. 5(5):913–960.
  • de Suzzoni, A.-S. (2013). Large data low regularity scattering results for the wave equation on the Euclidean space. Comm. Partial Differential Equations. 38(1):1–49.
  • de Suzzoni, A.-S. (2014). Consequences of the choice of a particular basis of L2(S3) for the cubic wave equation on the sphere and the Euclidean space. Commun. Pure Appl. Anal. 13(3):991–1015.
  • Lührmann, J., Mendelson, D. (2014). Random data Cauchy theory for nonlinear wave equations of power-type on R3. Comm. Partial Differential Equations. 39(12):2262–2283.
  • Lührmann, J., Mendelson, D. (2016). On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on R3. N. Y. J. Math. 22:209–227.
  • Murphy, J. Random data final-state problem for the mass-subcritical NLS in L2. Proc. Amer. Math. Soc. DOI: 10.1090/proc/14275
  • Poiret, A. (2012). Solutions globales pour l‘equation de Schrödinger cubique en dimension 3. Preprint arXiv:1207.1578.
  • Poiret, A. (2012). Solutions globales pour des equations de Schrödinger sur-critiques en toutes dimensions. Preprint arXiv:1207.3519.
  • Poiret, A., Robert, D., Thomann, L. (2014). Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. Anal. PDE. 7(4):997–1026.
  • Thomann, L. (2009). Random data Cauchy problem for supercritical Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire. 26(6):2385–2402.
  • Dodson, B., Lührmann, J., Mendelson, D. (2018). Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation. Preprint arXiv:1802.03795
  • Zhang, X. (2006). On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations. J. Differential Equations. 230(2):422–445.
  • Tao, T., Visan, M., Zhang, X. (2007). The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm. Partial Differential Equations. 32(7–9):1281–1343.
  • Killip, R., Visan, M., Zhang, X. (2008). The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE. 1(2):229–266.
  • Constantin, P., Saut, J.-C. (1988). Local smoothing properties of dispersive equations. J. Amer. Math. Soc. Soc. 1:413–439.
  • Sjölin, P. (1987). Regularity of solutions to the Schrödinger equation. Duke Math. J. 55:699–715.
  • Vega, L. (1988). Schrödinger equations: pointwise convergence to the initial data. Proc. Amer. Math. Soc. 102:874–878.

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