-
Previous Article
A shift map with a discontinuous entropy function
- DCDS Home
- This Issue
-
Next Article
Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux
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 |
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.
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 Lp 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 Lp 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]. |
[1] |
Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382 |
[2] |
Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 |
[3] |
G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 |
[4] |
Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure and Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727 |
[5] |
Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605 |
[6] |
Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863 |
[7] |
Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 |
[8] |
Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149 |
[9] |
Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253 |
[10] |
Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565 |
[11] |
Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146 |
[12] |
Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 |
[13] |
M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573 |
[14] |
Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029 |
[15] |
Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673 |
[16] |
Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139 |
[17] |
Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803 |
[18] |
Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082 |
[19] |
Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1 |
[20] |
Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]