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On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs

  • Árpád BényiEmail author
  • Tadahiro Oh
  • Oana Pocovnicu
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.

Notes

Acknowledgements

Á. B. is partially supported by a grant from the Simons Foundation (No. 246024). T. O. was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The authors would like to thank Justin Forlano for careful proofreading.

References

  1. 1.
    S. Albeverio, A. Cruzeiro, Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids, Comm. Math. Phys. 129 (1990) 431–444.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S. Albeverio, S. Kusuoka The invariant measure and the flow associated to the \(\Phi ^4_3\)-quantum field model, arXiv:1711.07108 [math.PR].
  3. 3.
    A. Ayache, N. Tzvetkov, \(L^p\)properties for Gaussian random series, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4425–4439.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Á. Bényi, T. Oh, Modulation spaces, Wiener amalgam spaces, and Brownian motions, Adv. Math. 228 (2011), no. 5, 2943–2981.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Á. Bényi, T. Oh, 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.Google Scholar
  6. 6.
    Á. Bényi, T. Oh, O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on \(\mathbb{R}^3\), \(d\ge 3\), Trans. Amer. Math. Soc. Ser. B 2 (2015), 1–50.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Á. Bényi, T. Oh, 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.Google Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys. 166 (1994), no. 1, 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    J. Bourgain, Invariant measures for the \(2D\)-defocusing nonlinear Schrödinger equation, Comm. Math. Phys. 176 (1996), no. 2, 421–445.Google Scholar
  11. 11.
    J. Bourgain, Invariant measures for the Gross-Piatevskii equation, J. Math. Pures Appl. 76 (1997), no. 8, 649–702.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices 1998, no. 5, 253–283.zbMATHCrossRefGoogle Scholar
  13. 13.
    J. Bourgain, A. Bulut, Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, J. Funct. Anal. 266 (2014), no. 4, 2319–2340.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    J. Bourgain, A. Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: the 2D case, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 6, 1267–1288.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Bourgain, A. Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 6, 1289–1325.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    J. Bourgain, C. Demeter, The proof of the \(l^2\)decoupling conjecture, Ann. of Math. 182 (2015), no. 1, 351–389.Google Scholar
  17. 17.
    H. Brezis, T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996), 277–304.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    N. Burq, L. Thomann, N. Tzvetkov, Long time dynamics for the one dimensional non linear Schrödinger equation, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2137–2198.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    N. Burq, L. Thomann, N. Tzvetkov, Global infinite energy solutions for the cubic wave equation, Bull. Soc. Math. France 143 (2015), no. 2, 301–313.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    N. Burq, L. Thomann, N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, Ann. Fac. Sci. Toulouse Math. 27 (2018), no. 3, 527–597.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 (2008), no. 3, 449–475.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    N. Burq, N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 1, 1–30.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    R. Catellier, K. Chouk, Paracontrolled distributions and the 3-dimensional stochastic quantization equation, Ann. Probab. 46 (2018), no. 5, 2621–2679.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    A. Choffrut, O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Internat. Math. Res. Not. Volume 2018, no.3, 699–738.MathSciNetzbMATHGoogle Scholar
  25. 25.
    M. Christ, J. Colliander, T. Tao, Instability of the periodic nonlinear Schrödinger equation, arXiv:math/0311227v1 [math.AP].
  26. 26.
    M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv:math/0311048 [math.AP].
  27. 27.
    J. Colliander, T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below \(L^2(\mathbb{T} )\), Duke Math. J. 161 (2012), no. 3, 367–414.Google Scholar
  28. 28.
    G. Da Prato, A. Debussche, Two-dimensional Navier-Stokes equations driven by a space-time white noise, J. Funct. Anal. 196 (2002), no. 1, 180–210.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge, 2014. xviii+493 pp.Google Scholar
  30. 30.
    A. de Bouard, A. Debussche, The stochastic nonlinear Schrödinger equation in \(H^1\), Stochastic Anal. Appl. 21 (2003), no. 1, 97–126.Google Scholar
  31. 31.
    A.S. de Suzzoni, Large data low regularity scattering results for the wave equation on the Euclidean space, Comm. Partial Differential Equations 38 (2013), no. 1, 1–49.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    A.S. de Suzzoni, F. Cacciafesta, Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line, arXiv:1509.02093 [math.AP].
  33. 33.
    B. Dodson, J. Lührmann, D. Mendelson, Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data, arXiv:1703.09655 [math.AP].
  34. 34.
    B. Dodson, J. Lührmann, D. Mendelson, Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, arXiv:1802.03795 [math.AP].
  35. 35.
    H. Feichtinger, Modulation spaces of locally compact Abelian groups, Technical report, University of Vienna (1983). in Proc. Internat. Conf. on Wavelets and Applications (Chennai, 2002), R. Radha, M. Krishna, S. Thangavelu (eds.), New Delhi Allied Publishers (2003), 1–56.Google Scholar
  36. 36.
    H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Func. Anal. 86 (1989), 307–340.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), 129–148.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    J. Forlano, T. Oh, Y. Wang, Stochastic cubic nonlinear Schrödinger equation with almost space-time white noise, arXiv:1805.08413 [math.AP].
  39. 39.
    L. Gross, Abstract Wiener spaces, Proc. 5th Berkeley Sym. Math. Stat. Prob. 2 (1965), 31–42.Google Scholar
  40. 40.
    M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi 3 (2015), e6, 75 pp.Google Scholar
  41. 41.
    M. Gubinelli, H. Koch, T. Oh, Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7335–7359.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Z. Guo, T. Oh, Non-existence of solutions for the periodic cubic nonlinear Schrödinger equation below \(L^2\), Internat. Math. Res. Not. 2018, no. 6, 1656–1729.Google Scholar
  43. 43.
    Z. Guo, T. Oh, Y. Wang, Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc. 109 (2014), no. 4, 975–1013.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    M. Hairer, A theory of regularity structures, Invent. Math. 198 (2014), no. 2, 269–504.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    M. Hairer, K. Matetski, Discretisations of rough stochastic PDEs, Ann. Probab. 46 (2018), no. 3, 1651–1709.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    J.P. Kahane, Some Random Series of Functions, Second edition. Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. xiv+305 pp.Google Scholar
  47. 47.
    R. Killip, J. Murphy, M. Vişan, Almost sure scattering for the energy-critical NLS with radial data below \(H^1(\mathbb{R}^4)\), to appear in Comm. Partial Differential Equations.Google Scholar
  48. 48.
    N. Kishimoto, A remark on norm inflation for nonlinear Schrödinger equations, arXiv:1806.10066 [math.AP].
  49. 49.
    H. Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975. vi+224 pp.Google Scholar
  50. 50.
    A. Kupiainen, Renormalization group and stochastic PDEs, Ann. Henri Poincaré 17 (2016), no. 3, 497–535.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    H. Lindblad, C. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    J. Lührmann, D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on \(\mathbb{R}^3\), Comm. Partial Differential Equations 39 (2014), no. 12, 2262–2283.Google Scholar
  53. 53.
    J. Lührmann, 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.Google Scholar
  54. 54.
    H.P. McKean, Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 168 (1995), no. 3, 479–491. Erratum: Statistical mechanics of nonlinear wave equations. IV. Cubic Schrödinger, Comm. Math. Phys. 173 (1995), no. 3, 675.Google Scholar
  55. 55.
    Y. Meyer, Wavelets and operators, Translated from the 1990 French original by D. H. Salinger. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992. xvi+224 pp.Google Scholar
  56. 56.
    L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett. 16 (2009), no. 1, 111–120.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    A. Nahmod, N. Pavlović, G. Staffilani, Gigliola Almost sure existence of global weak solutions for supercritical Navier-Stokes equations, SIAM J. Math. Anal. 45 (2013), no. 6, 3431–3452.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    A. Nahmod, G. Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1687–1759.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    E. Nelson, A quartic interaction in two dimensions, 1966 Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) pp. 69–73 M.I.T. Press, Cambridge, Mass.Google Scholar
  60. 60.
    T. Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE 2 (2009), no. 3, 281–304.MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    T.  Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac. 54 (2011), no. 3, 335–365.MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    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.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    T. Oh, M. Okamoto, O. Pocovnicu, On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities, arXiv:1708.01568 [math.AP].
  64. 64.
    T. Oh, M. Okamoto, N. Tzvetkov, Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation, preprint.Google Scholar
  65. 65.
    T. Oh, 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.Google Scholar
  66. 66.
    T. Oh, O. Pocovnicu, A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting, Tohoku Math. J. 69 (2017), no.3, 455–481.MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    T. Oh, O. Pocovnicu, N. Tzvetkov, Probabilistic local Cauchy theory of the cubic nonlinear wave equation in negative Sobolev spaces, preprint.Google Scholar
  68. 68.
    T. Oh, O. Pocovnicu, Y. Wang, On the stochastic nonlinear Schrödinger equations with non-smooth additive noise, to appear in Kyoto J. Math.Google Scholar
  69. 69.
    T. Oh, J. Quastel, On Cameron-Martin theorem and almost sure global existence, Proc. Edinb. Math. Soc. 59 (2016), 483–501.MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    T. Oh, L. Thomann, A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations, Stoch. Partial Differ. Equ. Anal. Comput. 6 (2018), 397–445.CrossRefGoogle Scholar
  71. 71.
    T. Oh, L. Thomann, Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations, to appear in Ann. Fac. Sci. Toulouse Math.Google Scholar
  72. 72.
    T. Oh, N. Tzvetkov, Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Related Fields 169 (2017), 1121–1168.MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    T. Oh, N. Tzvetkov, On the transport of Gaussian measures under the flow of Hamiltonian PDEs, Sémin. Équ. Dériv. Partielles. 2015-2016, Exp. No. 6, 9 pp.Google Scholar
  74. 74.
    T. Oh, N. Tzvetkov, Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation, arXiv:1703.10718 [math.AP].
  75. 75.
    T. Oh, N. Tzvetkov, Y. Wang, Solving the 4NLS with white noise initial data, preprint.Google Scholar
  76. 76.
    T. Oh, Y. Wang, On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle, to appear in An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.).Google Scholar
  77. 77.
    T. Ozawa, Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201–222.MathSciNetzbMATHGoogle Scholar
  78. 78.
    R.E.A.C. Paley, A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Philos. Soc. 26 (1930), 337–357, 458–474; 28 (1932), 190–205.Google Scholar
  79. 79.
    O. Pocovnicu, Probabilistic global well-posedness of the energy-critical defocusing cubic nonlinear wave equations on \(\mathbb{R}^4\), J. Eur. Math. Soc. (JEMS) 19 (2017), 2321–2375.MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    O. Pocovnicu, Y. Wang, An \(L^p\)-theory for almost sure local well-posedness of the nonlinear Schrödinger equations, preprint.Google Scholar
  81. 81.
    A. Poiret, D. Robert, L. Thomann, Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator, Anal. PDE 7 (2014), no. 4, 997–1026.MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    G. Richards, Invariance of the Gibbs measure for the periodic quartic gKdV, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 3, 699–766.MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    B. Simon, The \(P(\varphi )_2\)Euclidean (quantum) field theory, Princeton Series in Physics. Princeton University Press, Princeton, N.J., 1974. xx+392 pp.Google Scholar
  84. 84.
    C. Sun, B. Xia, Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three, Illinois J. Math. 60 (2016), no. 2, 481–503.MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp.Google Scholar
  86. 86.
    L. Thomann, N. Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity 23 (2010), no. 11, 2771–2791.MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    N. Tzvetkov, Quasiinvariant Gaussian measures for one-dimensional Hamiltonian partial differential equations, Forum Math. Sigma 3 (2015), e28, 35 pp.Google Scholar
  88. 88.
    N. Tzvetkov, Random data wave equations, arXiv:1704.01191 [math.AP].
  89. 89.
    N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100.Google Scholar
  90. 90.
    B. Xia, Generic ill-posedness for wave equation of power type on 3D torus, arXiv:1507.07179 [math.AP].
  91. 91.
    V. Yudovich, Non-stationary flows of an ideal incompressible fluid, Zh. Vychisl. Math. i Math. Fiz. (1963) 1032–1066 (in Russian).Google Scholar
  92. 92.
    T. Zhang, D. Fang, Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech. 14 (2012), no. 2, 311–324.MathSciNetzbMATHCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsWestern Washington UniversityBellinghamUSA
  2. 2.School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s BuildingsEdinburghUK
  3. 3.Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical SciencesEdinburghUK

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