# American Institute of Mathematical Sciences

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 [3] 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.  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. Hadac, S. 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. Kishimoto, M. 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

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. 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.  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. Hadac, S. 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. Kishimoto, M. 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
 [1] Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2020393 [2] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006 [3] Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 [4] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [5] Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015 [6] Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021001 [7] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [8] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [9] Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002 [10] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [11] Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 [12] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [13] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [14] Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 [15] Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024 [16] Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299 [17] Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139 [18] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [19] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [20] Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287

2019 Impact Factor: 1.338