March  2020, 40(3): 1283-1307. doi: 10.3934/dcds.2020078

Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space

1. 

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

Received  September 2018 Revised  August 2019 Published  December 2019

Fund Project: The first author N.K is partially supported by JSPS KAKENHI Grant-in-Aid for Young Researchers (B) (16K17626). The second author M.S is supported by China Postdoctoral Science Foundation grant 2019M650872. The third author Y.T is partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) (17H02853) and Grant-in-Aid for Exploratory Research (16K13770).

The global well-posedness for the KP-Ⅱ equation is established in the anisotropic Sobolev space $ H^{s, 0} $ for $ s>-\frac{3}{8} $. Even though conservation laws are invalid in the Sobolev space with negative index, we explore the asymptotic behavior of the solution by the aid of the $ I $-method in which Colliander, Keel, Staffilani, Takaoka, and Tao introduced a series of modified energy terms. Moreover, a-priori estimate of the solution leads to the existence of global attractor for the weakly damped, forced KP-Ⅱ equation in the weak topology of the Sobolev space when $ s>-\frac{1}{8} $.

Citation: Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[2]

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

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J. CollianderM. KeelG. StaffilaniH. 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.  Google Scholar

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M. Hadac, Well-posedness for the Kadomtsev-Petviashvili Ⅱ equation and generalisations, Trans. Amer. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.  Google Scholar

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M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

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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 Differential Equations, 6 (2001), 1027-1054.  doi: 10.1081/PDE-100002387.  Google Scholar

[7]

P. Isaza and J. Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differential Equations, (2003), 12pp.  Google Scholar

[8]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541.   Google Scholar

[9]

C. E. Kenig and S. N. Ziesler, Local well posedness for modified Kadomstev-Petviashvili equations, Differential Integral Equations, 18 (2005), 1111-1146.   Google Scholar

[10]

N. KishimotoM. Shan and Y. Tsutsumi, Localization estimate and global attractor for the damped and forced Zakharov-Kuznetsov equation in $\mathbb {R}^2$, Dyn. Partial Differ. Equ., 16 (2019), 317-323.  doi: 10.4310/DPDE.2019.v16.n4.a1.  Google Scholar

[11]

H. Koch and D. Tataru, Dispersive estimates for principlally normal pseudo-differential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

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H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, (2007), 36–71. doi: 10.1093/imrn/rnm053.  Google Scholar

[13]

H. Koch and D. Tataru, Energy and local energy bounds for the 1D cubic NLS equation in $H^{\frac{1}{4}}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988.  doi: 10.1016/j.anihpc.2012.05.006.  Google Scholar

[14]

A. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on $\mathbb {R}^n$, J. Differential Equations, 132 (1996), 87-106.  doi: 10.1006/jdeq.1996.0172.  Google Scholar

[15]

G. Prashant, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Preprint, 2018. Google Scholar

[16]

J. C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.  doi: 10.1512/iumj.1993.42.42047.  Google Scholar

[17]

E. I. Schul'man and V. E. Zakharov, Degenerative dispersion laws, motion invariants and kinetic equations, Phys. D., 1 (1980), 192-202.  doi: 10.1016/0167-2789(80)90011-1.  Google Scholar

[18]

M. J. Shan, Well-posedness for the two-dimensional Zakharov-Kuznetsov equation, preprint, arXiv: math/1807.10123. Google Scholar

[19]

M. J. Shan, Global well-posedness and global attractor for two-dimensional Zakharov-Kuznetsov equation, preprint, arXiv: math/1810.02984. Google Scholar

[20]

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

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[22]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

[23]

B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[24]

B. X. Wang, Z. H. Huo, C. C. Hao and Z. H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[2]

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

[3]

J. CollianderM. KeelG. StaffilaniH. 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.  Google Scholar

[4]

M. Hadac, Well-posedness for the Kadomtsev-Petviashvili Ⅱ equation and generalisations, Trans. Amer. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.  Google Scholar

[5]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar

[6]

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 Differential Equations, 6 (2001), 1027-1054.  doi: 10.1081/PDE-100002387.  Google Scholar

[7]

P. Isaza and J. Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differential Equations, (2003), 12pp.  Google Scholar

[8]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541.   Google Scholar

[9]

C. E. Kenig and S. N. Ziesler, Local well posedness for modified Kadomstev-Petviashvili equations, Differential Integral Equations, 18 (2005), 1111-1146.   Google Scholar

[10]

N. KishimotoM. Shan and Y. Tsutsumi, Localization estimate and global attractor for the damped and forced Zakharov-Kuznetsov equation in $\mathbb {R}^2$, Dyn. Partial Differ. Equ., 16 (2019), 317-323.  doi: 10.4310/DPDE.2019.v16.n4.a1.  Google Scholar

[11]

H. Koch and D. Tataru, Dispersive estimates for principlally normal pseudo-differential operators, Comm. Pure Appl. Math., 58 (2005), 217-284.  doi: 10.1002/cpa.20067.  Google Scholar

[12]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, (2007), 36–71. doi: 10.1093/imrn/rnm053.  Google Scholar

[13]

H. Koch and D. Tataru, Energy and local energy bounds for the 1D cubic NLS equation in $H^{\frac{1}{4}}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988.  doi: 10.1016/j.anihpc.2012.05.006.  Google Scholar

[14]

A. Merino, On the existence of the compact global attractor for semilinear reaction diffusion systems on $\mathbb {R}^n$, J. Differential Equations, 132 (1996), 87-106.  doi: 10.1006/jdeq.1996.0172.  Google Scholar

[15]

G. Prashant, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Preprint, 2018. Google Scholar

[16]

J. C. Saut, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42 (1993), 1011-1026.  doi: 10.1512/iumj.1993.42.42047.  Google Scholar

[17]

E. I. Schul'man and V. E. Zakharov, Degenerative dispersion laws, motion invariants and kinetic equations, Phys. D., 1 (1980), 192-202.  doi: 10.1016/0167-2789(80)90011-1.  Google Scholar

[18]

M. J. Shan, Well-posedness for the two-dimensional Zakharov-Kuznetsov equation, preprint, arXiv: math/1807.10123. Google Scholar

[19]

M. J. Shan, Global well-posedness and global attractor for two-dimensional Zakharov-Kuznetsov equation, preprint, arXiv: math/1810.02984. Google Scholar

[20]

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

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[22]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.  Google Scholar

[23]

B. X. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128 (1999), 41-52.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[24]

B. X. Wang, Z. H. Huo, C. C. Hao and Z. H. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/9789814360746.  Google Scholar

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