Almost sure global well posedness for the BBM equation with infinite <inline-formula><tex-math id="M1">\begin{document} L^{2} \end{document}</tex-math></inline-formula> initial data

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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.

Keywords: BBM equation, almost sure local well-posedness, almost sure global well-posedness, ill-posedness, norm inflation, stability.

Mathematics Subject Classification: Primary: 35Q53, 76B15.

Citation: Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011

References:

[1]	A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166.
[2]	T. B. Benjamin, J. 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.
[3]	Á. Bényi, T. 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.
[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.
[5]	J. L. Bona, M. 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.
[6]	J. L. Bona, M. 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.
[7]	J. L. Bona, T. 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.
[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.
[9]	J. L. Bona, W. 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.
[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.
[11]	J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26. doi: 10.1007/BF02099299.
[12]	J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445. doi: 10.1007/BF02099556.
[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.
[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.
[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.
[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.
[17]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387.
[18]	J. Colliander, M. Keel, G. Staffilani, H. 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.
[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.
[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.
[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.
[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.
[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.
[24]	S. S. Dragomir, Some Gronwall Type Inequalities and Applications, , Nova Science Publishers, Inc., Hauppauge, NY, 2003.
[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.
[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.
[27]	J. Ginibre, Y. 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.
[28]	M. Gubinelli, H. 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.
[29]	M. Gubinelli, H. Koch, T. Oh and L. Tolomeo, Global dynamics for the two-dimensional stochastic nonlinear wave equations, preprint.
[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.
[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.
[32]	M. Hairer, An Introduction to Stochastic PDEs, Available from: http://www.hairer.org/notes/SPDEs.pdf, 2009.
[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.
[34]	S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.
[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.
[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.
[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.
[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.
[39]	T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac., 54 (2011), 335-365.
[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.
[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].
[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.
[43]	D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.
[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.
[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.
[46]	L. Tolomeo, Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain, preprint.
[47]	N. Tzvetkov, Random data wave equations, arXiv: 1704.01191 [math.AP].
[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.
[49]	M. Wang, Sharp global well-posedness of the BBM equation in L^p type Sobolev spaces, Discrete Contin. Dyn. Sys., 36 (2016), 5763-5788. doi: 10.3934/dcds.2016053.
[50]	B. Xia, Generic ill-posedness for wave equation of power type on 3D torus, arXiv: 1507.07179 [math.AP].

show all references

References:

[1]	A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Adv. Differential Equations, 11 (2006), 121-166.
[2]	T. B. Benjamin, J. 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.
[3]	Á. Bényi, T. 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.
[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.
[5]	J. L. Bona, M. 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.
[6]	J. L. Bona, M. 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.
[7]	J. L. Bona, T. 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.
[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.
[9]	J. L. Bona, W. 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.
[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.
[11]	J. Bourgain, Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166 (1994), 1-26. doi: 10.1007/BF02099299.
[12]	J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176 (1996), 421-445. doi: 10.1007/BF02099556.
[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.
[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.
[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.
[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.
[17]	J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669. doi: 10.1137/S0036141001384387.
[18]	J. Colliander, M. Keel, G. Staffilani, H. 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.
[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.
[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.
[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.
[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.
[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.
[24]	S. S. Dragomir, Some Gronwall Type Inequalities and Applications, , Nova Science Publishers, Inc., Hauppauge, NY, 2003.
[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.
[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.
[27]	J. Ginibre, Y. 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.
[28]	M. Gubinelli, H. 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.
[29]	M. Gubinelli, H. Koch, T. Oh and L. Tolomeo, Global dynamics for the two-dimensional stochastic nonlinear wave equations, preprint.
[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.
[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.
[32]	M. Hairer, An Introduction to Stochastic PDEs, Available from: http://www.hairer.org/notes/SPDEs.pdf, 2009.
[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.
[34]	S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.
[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.
[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.
[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.
[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.
[39]	T. Oh, Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, Funkcial. Ekvac., 54 (2011), 335-365.
[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.
[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].
[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.
[43]	D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.
[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.
[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.
[46]	L. Tolomeo, Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain, preprint.
[47]	N. Tzvetkov, Random data wave equations, arXiv: 1704.01191 [math.AP].
[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.
[49]	M. Wang, Sharp global well-posedness of the BBM equation in L^p type Sobolev spaces, Discrete Contin. Dyn. Sys., 36 (2016), 5763-5788. doi: 10.3934/dcds.2016053.
[50]	B. Xia, Generic ill-posedness for wave equation of power type on 3D torus, arXiv: 1507.07179 [math.AP].

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