January  2020, 40(1): 267-318. doi: 10.3934/dcds.2020011

Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data

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

Received  January 2019 Revised  July 2019 Published  October 2019

Fund Project: J. F. was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh. J. F. also acknowledges support from Tadahiro Oh's ERC starting grant no. 637995 "ProbDynDispEq"

We consider the probabilistic Cauchy problem for the Benjamin-Bona-Mahony equation (BBM) on the one-dimensional torus $ \mathbb{T} $ with initial data below $ L^{2}( \mathbb{T}) $. With respect to random initial data of strictly negative Sobolev regularity, we prove that BBM is almost surely globally well-posed. The argument employs the $ I $-method to obtain an a priori bound on the growth of the 'residual' part of the solution. We then discuss the stability properties of the solution map in the deterministically ill-posed regime.

Citation: Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011
References:
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A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[3]

Á. BényiT. Oh and O. Pocovnicu, Higher order expansions of the probabilistic Cauchy theory of the cubic nonlinear Schödinger equation on $ \mathbb{R}^{3}$, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 114-160.  doi: 10.1090/btran/29.  Google Scholar

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

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J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small- amplitude long waves in nonlinear dispersive media. Ⅰ. Derivation and linear theory, J. Non- linear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

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J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

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J. L. BonaT. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

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J. L. Bona and M. Dai, Norm Inflation for the BBM equation, J. Math. Anal. Appl., 446 (2016), 879-885.  doi: 10.1016/j.jmaa.2016.08.067.  Google Scholar

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J. L. BonaW. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[10]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

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J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.  doi: 10.1007/BF02099299.  Google Scholar

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

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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.  Google Scholar

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N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. Ⅱ. A global existence result, Invent. Math., 173 (2008), 477-496.  doi: 10.1007/s00222-008-0123-0.  Google Scholar

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J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

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J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and Modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A.-S. de Suzzoni, Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures, Discrete Contin. Dyn. Syst., 35 (2015), 2905-2920.  doi: 10.3934/dcds.2015.35.2905.  Google Scholar

[22]

A.-S. de Suzzoni, Wave turbulence for the BBM equation: Stability of a Gaussian statistics under the flow of BBM, Comm. Math. Phys., 326 (2014), 773-813.  doi: 10.1007/s00220-014-1897-0.  Google Scholar

[23]

A.-S. de Suzzoni and N. Tzvetkov, On the propagation of weakly nonlinear random dispersive waves, Arch. Ration. Mech. Anal., 212 (2014), 849-874.  doi: 10.1007/s00205-014-0728-y.  Google Scholar

[24]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, , Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

[25]

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

[26] P. K. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications., Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
[27]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zahkharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[28]

M. GubinelliH. Koch and T. Oh, Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc., 370 (2018), 7335-7359.  doi: 10.1090/tran/7452.  Google Scholar

[29]

M. Gubinelli, H. Koch, T. Oh and L. Tolomeo, Global dynamics for the two-dimensional stochastic nonlinear wave equations, preprint. Google Scholar

[30]

M. Gubinelli and N. Perkowski, Probabilistic approach to the stochastic Burgers equation, Stochastic Partial Differential Equations and Related Fields, 515–527, Springer Proc. Math. Stat., 229, Springer, Cham, 2018.  Google Scholar

[31]

M. Gubinelli and N. Perkowski, Lectures on singular stochastic PDEs, Ensaios Matemáticos [Mathematical Surveys], Sociedade Brasileira de Matemática, Rio de Janeiro, 29 (2015), 89 pp.  Google Scholar

[32]

M. Hairer, An Introduction to Stochastic PDEs, Available from: http://www.hairer.org/notes/SPDEs.pdf, 2009. Google Scholar

[33]

T. Iwabuchi and T. Ogawa, Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., 367 (2015), 2613-2630.  doi: 10.1090/S0002-9947-2014-06000-5.  Google Scholar

[34] S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511526169.  Google Scholar
[35]

N. Kishimoto, A remark on norm inflation for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 18 (2019), 1375-1402.  doi: 10.3934/cpaa.2019067.  Google Scholar

[36]

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.  Google Scholar

[37]

E. Nelson, A quartic interaction in two dimensions, 1966 Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), 1966, 69–73, M.I.T. Press, Cambridge, Mass.  Google Scholar

[38]

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.   Google Scholar

[39]

T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac., 54 (2011), 335-365.   Google Scholar

[40]

T. Oh, M. Okamoto and N. Tzvetkov, Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation, preprint. Google Scholar

[41]

T. Oh, O. Pocovnicu and N. Tzvetkov, Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces, arXiv: 1904.06792 [math.AP]. Google Scholar

[42]

M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.  doi: 10.3934/dcds.2011.30.253.  Google Scholar

[43]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.   Google Scholar

[44]

D. Roumégoux, A symplectic non-squeezing theorem for BBM equation, Dyn. Partial Differ. Equ., 7 (2010), 289-305.  doi: 10.4310/DPDE.2010.v7.n4.a1.  Google Scholar

[45]

L. Thomann and N. Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 2771-2791.  doi: 10.1088/0951-7715/23/11/003.  Google Scholar

[46]

L. Tolomeo, Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain, preprint. Google Scholar

[47]

N. Tzvetkov, Random data wave equations, arXiv: 1704.01191 [math.AP]. Google Scholar

[48]

N. Tzvetkov, Quasi-invariant Gaussian measures for one dimensional Hamiltonian PDE's, Forum Math. Sigma, 3, (2015), e28, 35 pp. doi: 10.1017/fms.2015.27.  Google Scholar

[49]

M. Wang, Sharp global well-posedness of the BBM equation in Lp type Sobolev spaces, Discrete Contin. Dyn. Sys., 36 (2016), 5763-5788.  doi: 10.3934/dcds.2016053.  Google Scholar

[50]

B. Xia, Generic ill-posedness for wave equation of power type on 3D torus, arXiv: 1507.07179 [math.AP]. Google Scholar

show all references

References:
[1]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[3]

Á. BényiT. Oh and O. Pocovnicu, Higher order expansions of the probabilistic Cauchy theory of the cubic nonlinear Schödinger equation on $ \mathbb{R}^{3}$, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 114-160.  doi: 10.1090/btran/29.  Google Scholar

[4]

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

[5]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small- amplitude long waves in nonlinear dispersive media. Ⅰ. Derivation and linear theory, J. Non- linear Sci., 12 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.  Google Scholar

[6]

J. L. BonaM. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.  Google Scholar

[7]

J. L. BonaT. Colin and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

[8]

J. L. Bona and M. Dai, Norm Inflation for the BBM equation, J. Math. Anal. Appl., 446 (2016), 879-885.  doi: 10.1016/j.jmaa.2016.08.067.  Google Scholar

[9]

J. L. BonaW. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.  Google Scholar

[10]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.  doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[11]

J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26.  doi: 10.1007/BF02099299.  Google Scholar

[12]

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

[13]

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.  Google Scholar

[14]

N. Burq and N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. Ⅱ. A global existence result, Invent. Math., 173 (2008), 477-496.  doi: 10.1007/s00222-008-0123-0.  Google Scholar

[15]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not., 2018 (2018), 699-738.  doi: 10.1093/imrn/rnw246.  Google Scholar

[16]

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.  Google Scholar

[17]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[18]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and Modified KdV on $ \mathbb{R}$ and $ \mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[19]

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.  Google Scholar

[20]

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.  Google Scholar

[21]

A.-S. de Suzzoni, Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures, Discrete Contin. Dyn. Syst., 35 (2015), 2905-2920.  doi: 10.3934/dcds.2015.35.2905.  Google Scholar

[22]

A.-S. de Suzzoni, Wave turbulence for the BBM equation: Stability of a Gaussian statistics under the flow of BBM, Comm. Math. Phys., 326 (2014), 773-813.  doi: 10.1007/s00220-014-1897-0.  Google Scholar

[23]

A.-S. de Suzzoni and N. Tzvetkov, On the propagation of weakly nonlinear random dispersive waves, Arch. Ration. Mech. Anal., 212 (2014), 849-874.  doi: 10.1007/s00205-014-0728-y.  Google Scholar

[24]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, , Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

[25]

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

[26] P. K. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths. Theory and Applications., Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar
[27]

J. GinibreY. Tsutsumi and G. Velo, On the Cauchy problem for the Zahkharov system, J. Funct. Anal., 151 (1997), 384-436.  doi: 10.1006/jfan.1997.3148.  Google Scholar

[28]

M. GubinelliH. Koch and T. Oh, Renormalization of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc., 370 (2018), 7335-7359.  doi: 10.1090/tran/7452.  Google Scholar

[29]

M. Gubinelli, H. Koch, T. Oh and L. Tolomeo, Global dynamics for the two-dimensional stochastic nonlinear wave equations, preprint. Google Scholar

[30]

M. Gubinelli and N. Perkowski, Probabilistic approach to the stochastic Burgers equation, Stochastic Partial Differential Equations and Related Fields, 515–527, Springer Proc. Math. Stat., 229, Springer, Cham, 2018.  Google Scholar

[31]

M. Gubinelli and N. Perkowski, Lectures on singular stochastic PDEs, Ensaios Matemáticos [Mathematical Surveys], Sociedade Brasileira de Matemática, Rio de Janeiro, 29 (2015), 89 pp.  Google Scholar

[32]

M. Hairer, An Introduction to Stochastic PDEs, Available from: http://www.hairer.org/notes/SPDEs.pdf, 2009. Google Scholar

[33]

T. Iwabuchi and T. Ogawa, Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., 367 (2015), 2613-2630.  doi: 10.1090/S0002-9947-2014-06000-5.  Google Scholar

[34] S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997.  doi: 10.1017/CBO9780511526169.  Google Scholar
[35]

N. Kishimoto, A remark on norm inflation for nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 18 (2019), 1375-1402.  doi: 10.3934/cpaa.2019067.  Google Scholar

[36]

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.  Google Scholar

[37]

E. Nelson, A quartic interaction in two dimensions, 1966 Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), 1966, 69–73, M.I.T. Press, Cambridge, Mass.  Google Scholar

[38]

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.   Google Scholar

[39]

T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac., 54 (2011), 335-365.   Google Scholar

[40]

T. Oh, M. Okamoto and N. Tzvetkov, Uniqueness and non-uniqueness of the Gaussian free field evolution under the two-dimensional Wick ordered cubic wave equation, preprint. Google Scholar

[41]

T. Oh, O. Pocovnicu and N. Tzvetkov, Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces, arXiv: 1904.06792 [math.AP]. Google Scholar

[42]

M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.  doi: 10.3934/dcds.2011.30.253.  Google Scholar

[43]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.   Google Scholar

[44]

D. Roumégoux, A symplectic non-squeezing theorem for BBM equation, Dyn. Partial Differ. Equ., 7 (2010), 289-305.  doi: 10.4310/DPDE.2010.v7.n4.a1.  Google Scholar

[45]

L. Thomann and N. Tzvetkov, Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 2771-2791.  doi: 10.1088/0951-7715/23/11/003.  Google Scholar

[46]

L. Tolomeo, Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain, preprint. Google Scholar

[47]

N. Tzvetkov, Random data wave equations, arXiv: 1704.01191 [math.AP]. Google Scholar

[48]

N. Tzvetkov, Quasi-invariant Gaussian measures for one dimensional Hamiltonian PDE's, Forum Math. Sigma, 3, (2015), e28, 35 pp. doi: 10.1017/fms.2015.27.  Google Scholar

[49]

M. Wang, Sharp global well-posedness of the BBM equation in Lp type Sobolev spaces, Discrete Contin. Dyn. Sys., 36 (2016), 5763-5788.  doi: 10.3934/dcds.2016053.  Google Scholar

[50]

B. Xia, Generic ill-posedness for wave equation of power type on 3D torus, arXiv: 1507.07179 [math.AP]. Google Scholar

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