June  2019, 39(6): 3479-3520. doi: 10.3934/dcds.2019144

On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities

1. 

School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

2. 

Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan

3. 

Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom

* Corresponding author: Tadahiro Oh

Received  September 2018 Published  February 2019

Fund Project: The first author was supported by the European Research Council (grant no. 637995 "ProbDynDispEq"). The second author was supported by JSPS KAKENHI Grant number JP16K17624.

We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on $ \mathbb{R}^d $, $ d = 5,6 $, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (ⅰ) we prove almost sure global well-posedness of the (standard) energy-critical NLS on $ \mathbb{R}^d $, $ d = 5, 6 $, in the defocusing case, and (ⅱ) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space.

Citation: Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144
References:
[1]

A. Ayache and N. Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc., 360 (2008), 4425-4439.  doi: 10.1090/S0002-9947-08-04456-5.

[2]

Á. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in Harmonic Analysis, Vol. 4, 3–25, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2015.

[3]

Á. BényiT. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb{R} ^d$, $d \ge 3$, Trans. Amer. Math. Soc. Ser. B, 2 (2015), 1-50.  doi: 10.1090/btran/6.

[4]

Á. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory for nonlinear dispersive PDEs, Landscapes of Time-Frequency Analysis, 1–32, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2019.

[5]

Á. Bényi, T. Oh and O. Pocovnicu, Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb{R}^3$, to appear in Trans. Amer. Math. Soc.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[7]

J. Bourgain, Invariant measures for the 2$D$-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.  doi: 10.1007/BF02099556.

[8]

J. Bourgain, Refinements of Strichartz' inequality and applications to 2$D$-NLS with critical nonlinearity, Internat. Math. Res. Notices, 1998, 253–283. doi: 10.1155/S1073792898000191.

[9]

J. Brereton, Almost sure local well-posedness for the supercritical quintic NLS, Tunisian J. Math., 1 (2019), 427-453.  doi: 10.2140/tunis.2019.1.427.

[10]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations I: Local theory, Invent. Math., 173 (2008), 449-475.  doi: 10.1007/s00222-008-0124-z.

[11]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), 1-30.  doi: 10.4171/JEMS/426.

[12]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. doi: 10.1090/cln/010.

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J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\mathbb{T})$, Duke Math. J., 161 (2012), 367-414.  doi: 10.1215/00127094-1507400.

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G. Da Prato and A. Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal., 196 (2002), 180-210.  doi: 10.1006/jfan.2002.3919.

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G. Da Prato and A. Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab., 31 (2003), 1900-1916.  doi: 10.1214/aop/1068646370.

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C. Deng and S. Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\mathbb{T} ^3$, J. Differential Equations, 251 (2011), 902-917.  doi: 10.1016/j.jde.2011.05.002.

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P. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext. Springer, Cham, 2014. xiv+251 pp. doi: 10.1007/978-3-319-08332-2.

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J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.  doi: 10.1007/BF02099195.

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M. Gubinelli and N. Perkowski, Lectures on Singular Stochastic PDEs, Ensaios Matemáticos [Mathematical Surveys], 29. Sociedade Brasileira de Matemática, Rio de Janeiro, 2015. 89 pp.

[24]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917–941; Erratum to Well-posedness and Scattering for the KP-Ⅱ Equation in a Critical Space, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006.

[25]

M. Hairer, Introduction to regularity structures, Braz. J. Probab. Stat., 29 (2015), 175-210.  doi: 10.1214/14-BJPS241.

[26]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1 (\mathbb{T}^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.

[27]

H. Hirayama and M. Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., 36 (2016), 6943-6974.  doi: 10.3934/dcds.2016102.

[28]

M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.  doi: 10.1016/j.jmaa.2015.01.003.

[29]

M. Ikeda and T. Inui, Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.  doi: 10.1007/s00028-015-0273-7.

[30]

M. Ikeda and Y. Wakasugi, Small-data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285. 

[31]

A. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing NLS on $ \mathbb{R}\times \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.  doi: 10.1007/s00220-012-1474-3.

[32]

J. P. Kahane, Some Random Series of Functions, Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. xiv+305 pp.

[33]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[34]

R. Killip and M. Vişan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[35]

R. KillipT. OhO. Pocovnicu and M. Vişan, Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions, Math. Res. Lett., 19 (2012), 969-986.  doi: 10.4310/MRL.2012.v19.n5.a1.

[36]

R. Killip, J. Murphy and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb{R}^4)$, to appear in Comm. Partial Differential Equations.

[37]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.

[38]

J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $ \mathbb{R} ^3$, Comm. Partial Differential Equations, 39 (2014), 2262-2283.  doi: 10.1080/03605302.2014.933239.

[39]

J. Lührmann and D. Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $ \mathbb{R}^3$, New York J. Math., 22 (2016), 209-227. 

[40]

H. P. McKean, Statistical mechanics of nonlinear wave equations. Ⅳ. Cubic Schrödinger, Comm. Math. Phys., 168 (1995), 479–491. Erratum: "Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger", Comm. Math. Phys., 173 (1995), 675. doi: 10.1007/BF02101840.

[41]

T. Oh, A blowup result for the periodic NLS without gauge invariance, C. R. Math. Acad. Sci. Paris, 350 (2012), 389-392.  doi: 10.1016/j.crma.2012.04.009.

[42]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data, SIAM J. Math. Anal., 47 (2015), 1253-1270.  doi: 10.1137/140973384.

[43]

T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.  doi: 10.1619/fesi.60.259.

[44]

T. Oh and O. Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $ \mathbb{R}^3$, J. Math. Pures Appl., 105 (2016), 342-366.  doi: 10.1016/j.matpur.2015.11.003.

[45]

T. Oh and J. Quastel, On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc., 59 (2016), 483-501.  doi: 10.1017/S0013091515000218.

[46]

T. Oh, N. Tzvetkov and Y. Wang, Solving the 4NLS with white noise initial data, preprint.

[47]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222. 

[48]

R. E. A. C. Paley and A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Philos. Soc., 26 (1930), 337-357. 

[49]

O. Pocovnicu, Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $ \mathbb{R}^d$, $d = 4$ and $5$, J. Eur. Math. Soc. (JEMS), 19 (2017), 2521-2575.  doi: 10.4171/JEMS/723.

[50]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[51]

C. Sun and B. Xia, Probabilistic well-posedness for supercritical wave equation on $ \mathbb{T}^3$, Illinois J. Math., 60 (2016), 481-503. 

[52]

T. Tao and M. Vişan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations, 2005 (2005), 28 pp.

[53]

T. TaoM. Vişan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.

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M. Vişan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374.  doi: 10.1215/S0012-7094-07-13825-0.

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Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515-539.  doi: 10.1215/S0012-7094-99-09719-3.

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Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris, Ser. I, 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

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show all references

References:
[1]

A. Ayache and N. Tzvetkov, $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc., 360 (2008), 4425-4439.  doi: 10.1090/S0002-9947-08-04456-5.

[2]

Á. Bényi, T. Oh and O. Pocovnicu, Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS, Excursions in Harmonic Analysis, Vol. 4, 3–25, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2015.

[3]

Á. BényiT. Oh and O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb{R} ^d$, $d \ge 3$, Trans. Amer. Math. Soc. Ser. B, 2 (2015), 1-50.  doi: 10.1090/btran/6.

[4]

Á. Bényi, T. Oh and O. Pocovnicu, On the probabilistic Cauchy theory for nonlinear dispersive PDEs, Landscapes of Time-Frequency Analysis, 1–32, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2019.

[5]

Á. Bényi, T. Oh and O. Pocovnicu, Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb{R}^3$, to appear in Trans. Amer. Math. Soc.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Ⅰ. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[7]

J. Bourgain, Invariant measures for the 2$D$-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445.  doi: 10.1007/BF02099556.

[8]

J. Bourgain, Refinements of Strichartz' inequality and applications to 2$D$-NLS with critical nonlinearity, Internat. Math. Res. Notices, 1998, 253–283. doi: 10.1155/S1073792898000191.

[9]

J. Brereton, Almost sure local well-posedness for the supercritical quintic NLS, Tunisian J. Math., 1 (2019), 427-453.  doi: 10.2140/tunis.2019.1.427.

[10]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations I: Local theory, Invent. Math., 173 (2008), 449-475.  doi: 10.1007/s00222-008-0124-z.

[11]

N. Burq and N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS), 16 (2014), 1-30.  doi: 10.4171/JEMS/426.

[12]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp. doi: 10.1090/cln/010.

[13]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987), 18–29, Lecture Notes in Math., 1394, Springer, Berlin, 1989. doi: 10.1007/BFb0086749.

[14]

M. Christ, Power series solution of a nonlinear Schrödinger equation, Mathematical Aspects of Nonlinear Dispersive Equations, 131–155, Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, 2007.

[15]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv: 0311048, [math.AP].

[16]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb{R}^{3}$, Ann. of Math., (2) 167 (2008), 767–865. doi: 10.4007/annals.2008.167.767.

[17]

J. Colliander and T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^2(\mathbb{T})$, Duke Math. J., 161 (2012), 367-414.  doi: 10.1215/00127094-1507400.

[18]

G. Da Prato and A. Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal., 196 (2002), 180-210.  doi: 10.1006/jfan.2002.3919.

[19]

G. Da Prato and A. Debussche, Strong solutions to the stochastic quantization equations, Ann. Probab., 31 (2003), 1900-1916.  doi: 10.1214/aop/1068646370.

[20]

C. Deng and S. Cui, Random-data Cauchy problem for the Navier-Stokes equations on $\mathbb{T} ^3$, J. Differential Equations, 251 (2011), 902-917.  doi: 10.1016/j.jde.2011.05.002.

[21]

P. Friz and M. Hairer, A Course on Rough Paths. With an Introduction to Regularity Structures, Universitext. Springer, Cham, 2014. xiv+251 pp. doi: 10.1007/978-3-319-08332-2.

[22]

J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.  doi: 10.1007/BF02099195.

[23]

M. Gubinelli and N. Perkowski, Lectures on Singular Stochastic PDEs, Ensaios Matemáticos [Mathematical Surveys], 29. Sociedade Brasileira de Matemática, Rio de Janeiro, 2015. 89 pp.

[24]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917–941; Erratum to Well-posedness and Scattering for the KP-Ⅱ Equation in a Critical Space, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 971–972. doi: 10.1016/j.anihpc.2010.01.006.

[25]

M. Hairer, Introduction to regularity structures, Braz. J. Probab. Stat., 29 (2015), 175-210.  doi: 10.1214/14-BJPS241.

[26]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1 (\mathbb{T}^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.

[27]

H. Hirayama and M. Okamoto, Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity, Discrete Contin. Dyn. Syst., 36 (2016), 6943-6974.  doi: 10.3934/dcds.2016102.

[28]

M. Ikeda and T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance, J. Math. Anal. Appl., 425 (2015), 758-773.  doi: 10.1016/j.jmaa.2015.01.003.

[29]

M. Ikeda and T. Inui, Small data blow-up of $L^2$ or $H^1$-solution for the semilinear Schrödinger equation without gauge invariance, J. Evol. Equ., 15 (2015), 571-581.  doi: 10.1007/s00028-015-0273-7.

[30]

M. Ikeda and Y. Wakasugi, Small-data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance, Differential Integral Equations, 26 (2013), 1275-1285. 

[31]

A. Ionescu and B. Pausader, Global well-posedness of the energy-critical defocusing NLS on $ \mathbb{R}\times \mathbb{T}^3$, Comm. Math. Phys., 312 (2012), 781-831.  doi: 10.1007/s00220-012-1474-3.

[32]

J. P. Kahane, Some Random Series of Functions, Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. xiv+305 pp.

[33]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.

[34]

R. Killip and M. Vişan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424.  doi: 10.1353/ajm.0.0107.

[35]

R. KillipT. OhO. Pocovnicu and M. Vişan, Global well-posedness of the Gross-Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions, Math. Res. Lett., 19 (2012), 969-986.  doi: 10.4310/MRL.2012.v19.n5.a1.

[36]

R. Killip, J. Murphy and M. Visan, Almost sure scattering for the energy-critical NLS with radial data below $H^1(\mathbb{R}^4)$, to appear in Comm. Partial Differential Equations.

[37]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), Art. ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.

[38]

J. Lührmann and D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on $ \mathbb{R} ^3$, Comm. Partial Differential Equations, 39 (2014), 2262-2283.  doi: 10.1080/03605302.2014.933239.

[39]

J. Lührmann and D. Mendelson, On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on $ \mathbb{R}^3$, New York J. Math., 22 (2016), 209-227. 

[40]

H. P. McKean, Statistical mechanics of nonlinear wave equations. Ⅳ. Cubic Schrödinger, Comm. Math. Phys., 168 (1995), 479–491. Erratum: "Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger", Comm. Math. Phys., 173 (1995), 675. doi: 10.1007/BF02101840.

[41]

T. Oh, A blowup result for the periodic NLS without gauge invariance, C. R. Math. Acad. Sci. Paris, 350 (2012), 389-392.  doi: 10.1016/j.crma.2012.04.009.

[42]

T. Oh, On nonlinear Schrödinger equations with almost periodic initial data, SIAM J. Math. Anal., 47 (2015), 1253-1270.  doi: 10.1137/140973384.

[43]

T. Oh, A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.  doi: 10.1619/fesi.60.259.

[44]

T. Oh and O. Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on $ \mathbb{R}^3$, J. Math. Pures Appl., 105 (2016), 342-366.  doi: 10.1016/j.matpur.2015.11.003.

[45]

T. Oh and J. Quastel, On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc., 59 (2016), 483-501.  doi: 10.1017/S0013091515000218.

[46]

T. Oh, N. Tzvetkov and Y. Wang, Solving the 4NLS with white noise initial data, preprint.

[47]

T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 (1998), 201-222. 

[48]

R. E. A. C. Paley and A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Philos. Soc., 26 (1930), 337-357. 

[49]

O. Pocovnicu, Almost sure global well-posedness for the energy-critical defocusing nonlinear wave equation on $ \mathbb{R}^d$, $d = 4$ and $5$, J. Eur. Math. Soc. (JEMS), 19 (2017), 2521-2575.  doi: 10.4171/JEMS/723.

[50]

R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1.

[51]

C. Sun and B. Xia, Probabilistic well-posedness for supercritical wave equation on $ \mathbb{T}^3$, Illinois J. Math., 60 (2016), 481-503. 

[52]

T. Tao and M. Vişan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations, 2005 (2005), 28 pp.

[53]

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