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

Justin Forlano

doi:10.3934/dcds.2020011

Article Contents

2020, Volume 40, Issue 1: 267-318. Doi: 10.3934/dcds.2020011

This issue Previous Article Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux Next Article A shift map with a discontinuous entropy function

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

Justin Forlano

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

Received: December 31, 2018

Revised: June 30, 2019

Published: October 2019

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"

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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].