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Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces
1. | School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan 453007, China |
2. | Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, Hubei 430070, China |
3. | School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China |
4. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA |
$ \begin{align*} \partial_{x}\left(u_{t}-\beta\partial_{x}^{3}u +\partial_{x}(u^{2})\right)+\partial_{y}^{2}u-\gamma u = 0 \end{align*} $ |
$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $ |
$ \beta <0 $ |
$ \gamma >0, $ |
$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $ |
$ s_{1}>-\frac{1}{2} $ |
$ s_{2}\geq 0 $ |
$ H^{s_{1},0}(\mathbb{R}^{2}) $ |
$ s_{1}<-\frac{1}{2} $ |
$ C^{3} $ |
$ \beta <0,\gamma >0, $ |
$ U^{p} $ |
$ V^{p} $ |
$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $ |
References:
[1] |
L. A. Abramyan and Y. A. Stepanyants,
The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP., 61 (1985), 963-966.
|
[2] |
M. Ben-Artzi and J. C. Saut,
Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eqns., 12 (1999), 137-145.
|
[3] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[4] |
J. Bourgain,
On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[5] |
R. M. Chen, V. Hur and Y. Liu,
Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-2979.
doi: 10.1088/0951-7715/21/12/012. |
[6] |
R. M. Chen, Y. Liu and P. Z. Zhang,
Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Ameri. Math. Soc., 364 (2012), 3395-3425.
doi: 10.1090/S0002-9947-2012-05383-9. |
[7] |
J. Colliander, C. E. Kenig and G. Staffilani,
Low regularity solutions for the Kadomtsev-Petviashvili-I equation, Geom. Funct. Anal., 13 (2003), 737-794.
doi: 10.1007/s00039-003-0429-4. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eqns., 26 (2001), 7 pp. |
[10] |
J. Colliander, A. D. Ionescu, C. E. Kenig and G. Staffilani,
Weighted low-regularity solutions of the KP-I initial-value problem, Discrete Contin. Dyn. Syst., 20 (2008), 219-258.
doi: 10.3934/dcds.2008.20.219. |
[11] |
A. Esfahani and S. Levandosky,
Stability of solitary waves of the Kadomtsev–Petviashvili equation with a weak rotation, SIAM J. Math. Anal., 49 (2017), 5096-5133.
doi: 10.1137/16M1103865. |
[12] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, Séminaire Bourbaki, 1994/95 (1996), 163-187.
|
[13] |
R. Grimshaw,
Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stu. Appl. Math., 73 (1985), 1-33.
doi: 10.1002/sapm19857311. |
[14] |
R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira and Yu. A. Stepanyants,
Long nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophysics, 19 (1998), 289-338.
|
[15] |
A. Grünrock, New Applications of the Fourier Restriction Norm Method to Wellposedness Problems for Nonlinear Evolution Equations, Ph. D thesis, Universit$\ddot{a}$t Wuppertal in Dissertation, Germany, 2002. |
[16] |
Z. H. Guo, L. Z. Peng and B. X. Wang,
On the local regularity of the KP-I equation in anisotropic Sobolev space, J. Math. Pures Appl., 94 (2010), 414-432.
doi: 10.1016/j.matpur.2010.03.012. |
[17] |
M. Hadac,
Well-posedness for the Kadomtsev-Petviashvili II equation and generalizations, Trans. Ameri. Math. Soc., 360 (2008), 6555-6572.
doi: 10.1090/S0002-9947-08-04515-7. |
[18] |
M. Hadac, S. Herr and H. Koch,
Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré-AN, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[19] |
A. D. Ionescu, C. E. Kenig and D. Tataru,
Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.
doi: 10.1007/s00222-008-0115-0. |
[20] |
P. Isaza and J. Mejía,
Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Diff. Eqns., 26 (2001), 1027-1054.
doi: 10.1081/PDE-100002387. |
[21] |
C. E. Kenig,
On the local and global well-posedness theory for the KP-I equation, Ann I. H. Poincaré-AN, 21 (2004), 827-838.
doi: 10.1016/j.anihpc.2003.12.002. |
[22] |
B. B. Kadomtsev and V. I. Petviashvili,
On the stability of solitary waves in weakly dispersive media, Soviet. Phys. Dokl., 15 (1970), 539-541.
|
[23] |
C. E. Kenig, G. Ponce and L. Vega,
The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
doi: 10.1215/S0012-7094-93-07101-3. |
[24] |
C. E. Kenig, G. Ponce and L. Vega,
A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[25] |
C. E. Kenig and S. N. Ziesler,
Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146.
|
[26] |
H. Koch and D. Tataru, Dispersive estimates for principally normal pseudo-differential operators, Commu. Pure. Appl. Math., 58 (2005), 217-284.
doi: 10.1002/cpa. 20067. |
[27] |
H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), article ID rnm053, 36pp.
doi: 10.1093/imrn/rnm053. |
[28] |
H. Koch and J. F. Li,
Global well-posedness and scattering for small data for the three-dimensional Kadomtsev-Petviashvili II equation, Commun. Partial Diff. Eqns., 42 (2017), 950-976.
doi: 10.1080/03605302.2017.1320410. |
[29] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.
doi: 10.1215/S0012-7094-02-11525-7. |
[30] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275.
doi: 10.1007/s00208-002-0338-0. |
[31] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-I equation on the background of a non localized solution, Comm. Math. Phys., 272 (2007), 775-810.
doi: 10.1007/s00220-007-0243-1. |
[32] |
L. C. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré-AN, 28 (2011), 653-676.
doi: 10.1016/j.anihpc.2011.04.004. |
[33] |
H. Takaoka,
Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dyn. Syst., 6 (2000), 483-499.
doi: 10.3934/dcds.2000.6.483. |
[34] |
H. Takaoka,
Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443.
|
[35] |
H. Takaoka and N. Tzvetkov,
On the local regularity of the Kadomtsev-Petviashvili-II equation, Int. Math. Res. Not., 2001 (2001), 77-114.
doi: 10.1155/S1073792801000058. |
[36] |
N. Tzvetkov,
On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.
doi: 10.1080/03605309908821468. |
[37] |
N. Tzvetkov,
Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320.
|
[38] |
N. Wiener, The quadratic variation of a function and its Fourier coefficients, in Collected Works with Commentaries. Volume II: Generalized Harmonic analysis and Tauberian Theory; Classical Harmonic and Complex Analysis, Mathematicians of Our Time, 15, The MIT Press, Cambridge, MA-London, 1979. |
[39] |
Y. Zhang,
Local well-posedness of KP-I initial value problem on torus in the Besov space, Comm. Partial Diff. Eqns., 41 (2016), 256-281.
doi: 10.1080/03605302.2015.1126733. |
show all references
References:
[1] |
L. A. Abramyan and Y. A. Stepanyants,
The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP., 61 (1985), 963-966.
|
[2] |
M. Ben-Artzi and J. C. Saut,
Uniform decay estimates for a class of oscillatory integrals and applications, Diff. Int. Eqns., 12 (1999), 137-145.
|
[3] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
[4] |
J. Bourgain,
On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.
doi: 10.1007/BF01896259. |
[5] |
R. M. Chen, V. Hur and Y. Liu,
Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-2979.
doi: 10.1088/0951-7715/21/12/012. |
[6] |
R. M. Chen, Y. Liu and P. Z. Zhang,
Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Ameri. Math. Soc., 364 (2012), 3395-3425.
doi: 10.1090/S0002-9947-2012-05383-9. |
[7] |
J. Colliander, C. E. Kenig and G. Staffilani,
Low regularity solutions for the Kadomtsev-Petviashvili-I equation, Geom. Funct. Anal., 13 (2003), 737-794.
doi: 10.1007/s00039-003-0429-4. |
[8] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc., 16 (2003), 705-749.
doi: 10.1090/S0894-0347-03-00421-1. |
[9] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Diff. Eqns., 26 (2001), 7 pp. |
[10] |
J. Colliander, A. D. Ionescu, C. E. Kenig and G. Staffilani,
Weighted low-regularity solutions of the KP-I initial-value problem, Discrete Contin. Dyn. Syst., 20 (2008), 219-258.
doi: 10.3934/dcds.2008.20.219. |
[11] |
A. Esfahani and S. Levandosky,
Stability of solitary waves of the Kadomtsev–Petviashvili equation with a weak rotation, SIAM J. Math. Anal., 49 (2017), 5096-5133.
doi: 10.1137/16M1103865. |
[12] |
J. Ginibre,
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Astérisque, Séminaire Bourbaki, 1994/95 (1996), 163-187.
|
[13] |
R. Grimshaw,
Evolution equations for weakly nonlinear, long internal waves in a rotating fluid, Stu. Appl. Math., 73 (1985), 1-33.
doi: 10.1002/sapm19857311. |
[14] |
R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira and Yu. A. Stepanyants,
Long nonlinear surface and internal gravity waves in a rotating ocean, Surveys in Geophysics, 19 (1998), 289-338.
|
[15] |
A. Grünrock, New Applications of the Fourier Restriction Norm Method to Wellposedness Problems for Nonlinear Evolution Equations, Ph. D thesis, Universit$\ddot{a}$t Wuppertal in Dissertation, Germany, 2002. |
[16] |
Z. H. Guo, L. Z. Peng and B. X. Wang,
On the local regularity of the KP-I equation in anisotropic Sobolev space, J. Math. Pures Appl., 94 (2010), 414-432.
doi: 10.1016/j.matpur.2010.03.012. |
[17] |
M. Hadac,
Well-posedness for the Kadomtsev-Petviashvili II equation and generalizations, Trans. Ameri. Math. Soc., 360 (2008), 6555-6572.
doi: 10.1090/S0002-9947-08-04515-7. |
[18] |
M. Hadac, S. Herr and H. Koch,
Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré-AN, 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[19] |
A. D. Ionescu, C. E. Kenig and D. Tataru,
Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.
doi: 10.1007/s00222-008-0115-0. |
[20] |
P. Isaza and J. Mejía,
Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices, Comm. Partial Diff. Eqns., 26 (2001), 1027-1054.
doi: 10.1081/PDE-100002387. |
[21] |
C. E. Kenig,
On the local and global well-posedness theory for the KP-I equation, Ann I. H. Poincaré-AN, 21 (2004), 827-838.
doi: 10.1016/j.anihpc.2003.12.002. |
[22] |
B. B. Kadomtsev and V. I. Petviashvili,
On the stability of solitary waves in weakly dispersive media, Soviet. Phys. Dokl., 15 (1970), 539-541.
|
[23] |
C. E. Kenig, G. Ponce and L. Vega,
The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.
doi: 10.1215/S0012-7094-93-07101-3. |
[24] |
C. E. Kenig, G. Ponce and L. Vega,
A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.
doi: 10.1090/S0894-0347-96-00200-7. |
[25] |
C. E. Kenig and S. N. Ziesler,
Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146.
|
[26] |
H. Koch and D. Tataru, Dispersive estimates for principally normal pseudo-differential operators, Commu. Pure. Appl. Math., 58 (2005), 217-284.
doi: 10.1002/cpa. 20067. |
[27] |
H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not., 2007 (2007), article ID rnm053, 36pp.
doi: 10.1093/imrn/rnm053. |
[28] |
H. Koch and J. F. Li,
Global well-posedness and scattering for small data for the three-dimensional Kadomtsev-Petviashvili II equation, Commun. Partial Diff. Eqns., 42 (2017), 950-976.
doi: 10.1080/03605302.2017.1320410. |
[29] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115 (2002), 353-384.
doi: 10.1215/S0012-7094-02-11525-7. |
[30] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-I equation, Math. Ann., 324 (2002), 255-275.
doi: 10.1007/s00208-002-0338-0. |
[31] |
L. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-I equation on the background of a non localized solution, Comm. Math. Phys., 272 (2007), 775-810.
doi: 10.1007/s00220-007-0243-1. |
[32] |
L. C. Molinet, J. C. Saut and N. Tzvetkov,
Global well-posedness for the KP-II equation on the background of a non-localized solution, Ann. Inst. H. Poincaré-AN, 28 (2011), 653-676.
doi: 10.1016/j.anihpc.2011.04.004. |
[33] |
H. Takaoka,
Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dyn. Syst., 6 (2000), 483-499.
doi: 10.3934/dcds.2000.6.483. |
[34] |
H. Takaoka,
Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443.
|
[35] |
H. Takaoka and N. Tzvetkov,
On the local regularity of the Kadomtsev-Petviashvili-II equation, Int. Math. Res. Not., 2001 (2001), 77-114.
doi: 10.1155/S1073792801000058. |
[36] |
N. Tzvetkov,
On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.
doi: 10.1080/03605309908821468. |
[37] |
N. Tzvetkov,
Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320.
|
[38] |
N. Wiener, The quadratic variation of a function and its Fourier coefficients, in Collected Works with Commentaries. Volume II: Generalized Harmonic analysis and Tauberian Theory; Classical Harmonic and Complex Analysis, Mathematicians of Our Time, 15, The MIT Press, Cambridge, MA-London, 1979. |
[39] |
Y. Zhang,
Local well-posedness of KP-I initial value problem on torus in the Besov space, Comm. Partial Diff. Eqns., 41 (2016), 256-281.
doi: 10.1080/03605302.2015.1126733. |
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