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December  2021, 41(12): 5825-5849. doi: 10.3934/dcds.2021097

Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces

Wei Yan 1, , Yimin Zhang 2,, , Yongsheng Li 3, and  Jinqiao Duan 4,

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

* Corresponding author: Yimin Zhang

Received  February 2021 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: This work is supported by the NSFC under grant numbers 11771127, 11571118 and 11471330. The first author is also supported by the education department of Henan province under the grant number 21A110014. The third author is also supported by the Fundamental Research Funds for the Central Universities of China under the grant number 2017ZD094

We consider the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili (RMKP) equation
$ \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*} $
in the anisotropic Sobolev spaces
$ H^{s_{1},s_{2}}(\mathbb{R}^{2}) $
. When
$ \beta <0 $
 and
$ \gamma >0, $
 we prove that the Cauchy problem is locally well-posed in
$ H^{s_{1}, s_{2}}(\mathbb{R}^{2}) $
 with
$ s_{1}>-\frac{1}{2} $
 and
$ s_{2}\geq 0 $
. Our result considerably improves the Theorem 1.4 of R. M. Chen, Y. Liu, P. Z. Zhang(Transactions of the American Mathematical Society, 364(2012), 3395–3425.). The key idea is that we divide the frequency space into regular region and singular region. We further prove that the Cauchy problem for RMKP equation is ill-posed in
$ H^{s_{1},0}(\mathbb{R}^{2}) $
 with
$ s_{1}<-\frac{1}{2} $
 in the sense that the flow map associated to the rotation-modified Kadomtsev-Petviashvili is not
$ C^{3} $
. When
$ \beta <0,\gamma >0, $
 by using the
$ U^{p} $
 and
$ V^{p} $
 spaces, we prove that the Cauchy problem is locally well-posed in
$ H^{-\frac{1}{2},0}(\mathbb{R}^{2}) $
.
Keywords: Rotation modified KP equation, Cauchy problem, anisotropic Sobolev spaces.
Mathematics Subject Classification: Primary: 35Q53;Secondary: 35B30.
Citation: Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097
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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

C. E. Kenig and S. N. Ziesler, Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Takaoka, Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443.   Google Scholar

[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.  Google Scholar

[36]

N. Tzvetkov, On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.  doi: 10.1080/03605309908821468.  Google Scholar

[37]

N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

C. E. Kenig and S. N. Ziesler, Local well-posedness for modified Kadomtsev-Petviashvili equations, Diff. Int. Eqns., 10 (2005), 1111-1146.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Takaoka, Well-posedness for the Kadomtsev-Petviashvili II equation, Adv. Diff. Eqns., 5 (2000), 1421-1443.   Google Scholar

[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.  Google Scholar

[36]

N. Tzvetkov, On the Cauchy problem for Kadomtsev-Petviashvili equation, Comm. Partial Diff. Eqns., 24 (1999), 1367-1397.  doi: 10.1080/03605309908821468.  Google Scholar

[37]

N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Diff. Int. Eqns., 13 (2000), 1289-1320.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

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