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Randomness and Nonlinear Evolution Equations

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Abstract

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.

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References

  1. Albeverio, S., Cruzeiro, A. B.: Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids. Comm. Math. Phys., 129(3), 431–444 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Figalli, A.: Of flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions. J. Functional Analysis, 256, 179–214 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bejenaru, I., Ionescu, A., Kenig, C.: Global existence and uniqueness of Schrödinger maps in dimensions d ≥ 4. Adv. Math., 215(1), 263–291 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bejenaru, I., Ionescu, A., Kenig, C., et al.: Global Schrödinger maps in dimensions d ≥ 2: small data in the critical Sobolev spaces. Ann. of Math. (2), 173(3), 144–1506 (2011)

    Article  MATH  Google Scholar 

  5. Bényi, A., Oh, T., Pocovnicu, O.: On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝd, d ≥ 3. Trans. Amer. Math. Soc. Ser., B2, 1–50 (2015)

    MATH  Google Scholar 

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

    Google Scholar 

  7. Bogachev, V. I.: Gaussian Measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998

  8. Bourgain, J.: Periodic nonlinear Schrödinger equation and invariant measures. Commun. Math. Phys., 166, 1–26, 1994

    Article  MATH  Google Scholar 

  9. Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Comm. Math. Phys., 176(2), 421–445 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Buckdahn, R.: Anticipative Girsanov transformations. Probab. Th. Rel. Fields, 89, 211–238 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Buckmaster, T., Nahmod, A., Staffilani, G. et al.: The surface quasi-geostrophic equation with random diffusion, arXiv:1806.03734 [math.AP], 2018

    Google Scholar 

  12. Burq, N., Thomann, L., Tzvetkov, N.: Long time dynamics for the one dimensional non linear Schrödinger equation. Ann. Inst. Fourier (Grenoble), 63(6), 2137–2198 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Burq, N., Thomann, L., Tzvetkov, N.: Remarks on the Gibbs measures for nonlinear dispersive equations, arXiv:1412.7499, 2014

    Google Scholar 

  14. Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations I. Local theory. Invent. Math., 173(3), 449–475, (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Burq, N., Tzvetkov, N.: Random data Cauchy theory for supercritical wave equations. II. A global existence result. Invent. Math., 173(3), 477–496 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cambronero, S. H., McKean, H.: The ground state eigenvalue of Hills equation with white noise potential. Comm. Pure and Applied Mathematics, LII, 1277–1294 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cameron, R. H., Martin, W. T.: Transformation of Wiener integrals under translations. Ann. of Math., 45(2), 386–396 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  18. Colliander, J., Keel, M., Staffilani, G., et al.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3. Ann. of Math. (2), 167(3), 767–865 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Colliander, J., Oh, T.: Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2(T). Duke Math. J., 161(3), 367–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Constantin, P., Foias, C.: Navier-Stokes Equations, Chicago Lectures in Mathematics, The University of Chicago Press, 1988

    Google Scholar 

  21. Cote, R., Kenig, C., Lawrie, A., et al.: Characterization of large energy solutions of the equivariant wave map problem: I. Amer. J. Math., 137(1), 139–207 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Cote, R., Kenig, C., Lawrie, A., et al.: Characterization of large energy solutions of the equivariant wave map problem: II. Amer. J. Math., 137(1), 209–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cote, R., Kenig, C., Merle, F.: Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system. Comm. Math. Phys., 284(1), 203–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Deng, Y.: Two-dimensional nonlinear Schrödinger equation with random radial data. Anal. PDE, 5(5), 913–960 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dodson, B., L¨uhrmann, J., Mendelson, D.: Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation, arXiv:1802.03795, 2018

    Google Scholar 

  26. Duyckaerts, T., Kenig, C., Merle, F.: Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation. J. Eur. Math. Soc. (JEMS), 13(3), 533–599 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Duyckaerts, T., Kenig, C., Merle, F.: Profiles of bounded radial solutions of the focusing, energy-critical wave equation. Geom. Funct. Anal., 22(3), 639–698 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Duyckaerts, T., Kenig, C., Merle, F.: Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the nonradial case. J. Eur. Math. Soc. (JEMS), 14(5), 1389–1454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Enchev, O.: Nonlinear Transformations on the Wiener Space. The Annals of Probability, 21(4), 2169–2188 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Flandoli, F.: Random perturbation of PDEs and fluid dynamic models. 2015, Lecture Notes in Mathe-matics. Springer, Heidelberg, 2011. Lectures from the 40th Probability Summer School held in Saint-Flour, École d'Étéde Probabilités de Saint-Flour, 2010

    Google Scholar 

  31. Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Funct. Anal., 87(2), 359–369, (1989)

    Article  MathSciNet  MATH  Google Scholar 

  32. Glatt-Holtz, N., Vicol, V.: Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise. Ann. Probab., 42(1), 80–145 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Gr¨unrock, A., Herr, S.: Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data. SIAM J. Math. Anal., 39, 1890–1920 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Herr, S., Tataru, D., Tzvetkov, N.: Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H 1(T3). Duke Math. J., 159, 329–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ionescu, A. D., Pausader, B.: The energy-critical defocusing NLS on T3. Duke Math. J., 161(8), 1581–1612 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kallianpur, G., Karandikar, R. L.: Nonlinear transformations of the canonical Gauss measure on Hilbert space and absolute continuity. Acta Applicandae Mathematicae, 35, 63–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kenig, C.: Oscillatory integrals and nonlinear dispersive equations. Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), 268–285, Princeton Math. Ser., 42, Princeton Univ. Press, Princeton, NJ, 1995

    Google Scholar 

  38. Kenig, C.: The concentration-compactness rigidity method for critical dispersive and wave equations. Non-linear Partial Differential Equations, 117–149, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer Basel AG, Basel, 2012

    MATH  Google Scholar 

  39. Kenig, C., Merle, F.: 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 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing nonlinear wave equation. Acta Math., 201(2), 147–212 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kenig, C., Merle, F.: Scattering for H 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc., 362(4), 1937–1962 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kenig, C., Ponce, G., Rolvung, C., et al.: Variable coefficient Schrödinger flows for ultrahyperbolic operators. Adv. Math., 196(2), 373–486 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46(4), 527–620 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kenig, C., Ponce, G., Vega, L.: The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J., 71(1), 1–21 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kenig, C., Ponce, G., Vega, L.: Quadratic forms for the 1D semilinear Schrödinger equation. Trans. Amer. Math. Soc., 348, 3323–3353 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kenig, C., Ponce, G., Vega, L.: A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc., 9(2), 573–603 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kenig, C., Ponce, G., Vega, L.: Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math., 134(3), 489–545 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kenig, C., Ponce, G., Vega, L.: On the concentration of blow up solutions for the generalized KdV equation critical in L 2, Nonlinear wave equations (Providence, RI, 1998), 131–156, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000

    Google Scholar 

  49. Kenig, C., Ponce, G., Vega, L.: On the ill-posedness of some canonical dispersive equations. Duke Math. J., 106(3), 617–633, (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kenig, C., Ponce, G., Vega, L.: The Cauchy problem for quasi-linear Schrödinger equations. Invent. Math., 158(2), 343–388 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  51. Kusuoka, S.: The nonlinear transformation of Gaussian measure on Banach space and its absolute continuity (I). J. Fac. Univ. Tokyo Sect. IA Math., 29(3), 567–597 (1982)

    MathSciNet  MATH  Google Scholar 

  52. Lihrmann, J., Mendelson, D.: Random data Cauchy theory for nonlinear wave equations of power-type on ℝ3. Comm. Partial Differential Equations, 39(12), 2262–2283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lihrmann, J., Mendelson, D.: On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on R3. New York J. Math., 22, 209–227 (2016)

    MathSciNet  Google Scholar 

  54. Nahmod, A. R., Oh, T., Rey-Bellet, L., et al.: Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS. J. Eur. Math. Soc. (JEMS), 14(4), 1275–1330 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Nahmod, A. R., Pavlovic, N., Staffilani, G.: Almost sure existence of global weak solutions for super-critical Navier-Stokes equations. SIAM J. Math. Anal., 45(6), 3431–3452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  56. Nahmod, A. R., Pavlovic, N., Staffilani, G., et al.: Global flows with invariant measures for the inviscid modified SQG equations. Stoch. Partial Differ. Equ. Anal. Comput., 6(2), 184–210 (2018)

    MathSciNet  MATH  Google Scholar 

  57. Nahmod, A. R., Rey-Bellet, L., Sheffield, S., et al.: Absolute continuity of Brownian bridges under certain gauge transformations. Math. Res. Lett., 18(5), 875–887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  58. Nahmod, A. R., Staffilani, G.: Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space. JEMS, 17(7), 1687–1759 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Oh, T., Thomann, L.: A pedestrian approach to the invariant Gibbs measures for the 2-d de-focusing nonlinear Schrödinger equations. Stoch. Partial Differ. Equ. Anal. Comput., (2018)

    Google Scholar 

  60. Oksendal, B.: Stochastic Differential Equations, Springer-Verlag, Berlin, 2003

    Book  MATH  Google Scholar 

  61. Quastel, J., Valkó, B.: KdV preserves white noise. Communications in Mathematical Physics, 277(3), 707–714 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  62. Ramer, R.: On nonlinear transformations of Gaussian measures. J. Functional Anal., 15, 166–187 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  63. Thomann, L., Tzvetkov, N.: Gibbs measure for the periodic derivative nonlinear Schrödinger equation. Nonlinearity, 23(11), 2771–2791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Massachusetts Amherst, 710 N. Pleasant Street, Amherst, MA, 01003, USA

    Andrea R. Nahmod

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Gigliola Staffilani

Authors
  1. Andrea R. Nahmod
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  2. Gigliola Staffilani
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Correspondence to Andrea R. Nahmod.

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Dedicated to Carlos Kenig on his 65th birthday with gratitude and admiration

The first author was supported by NSF (Grant Nos. DMS 1201443 and DMS 1463714), the second author by NSF (Grant Nos. DMS 1362509 and DMS 1462401), the Simons Foundation and the John Simon Guggenheim Foundation

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Nahmod, A.R., Staffilani, G. Randomness and Nonlinear Evolution Equations. Acta. Math. Sin.-English Ser. 35, 903–932 (2019). https://doi.org/10.1007/s10114-019-8297-5

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