SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

Minjie JIANG; Wei YAN; Yimin ZHANG

doi:10.1016/S0252-9602(17)30058-9

Acta mathematica scientia, Series B >

2017 , Vol. 37 >Issue 4: 1061 - 1082

DOI: https://doi.org/10.1016/S0252-9602(17)30058-9

Articles

SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION

Minjie JIANG ,
Wei YAN ,
Yimin ZHANG

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1. College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China;
2. School of Science, Wuhan University of Technology, Wuhan 430070, China

Minjie JIANG,E-mail:1764915956@qq.com;Yimin ZHANG,E-mail:zhangym802@126.com

Received date: 2016-02-24

Revised date: 2016-12-24

Online published: 2017-08-25

Supported by

This work is supported by Natural Science Foundation of China NSFC (11401180 and 11471330). The second author is also supported by the Young Core Teachers Program of Henan Normal University (15A110033). The third author is also supported by the Fundamental Research Funds for the Central Universities (WUT:2017 IVA 075).

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Abstract

This current paper is devoted to the Cauchy problem for higher order dispersive equation
u_{t + ∂_x²ⁿ⁺¹u=∂_x(u∂_xⁿu) + ∂_x^n-1(u_x²), n ≥ 2, n ∈ N⁺.

By using Besov-type spaces, we prove that the associated problem is locally well-posed in H(-n/2 + 3/4,-1/2n)(R). The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^(s,a)(R) with s < -n/2 + 3/4 and all a ∈ R.}

Key words： Cauchy problem; sharp well-posedness; modified Bourgain spaces

Cite this article

Minjie JIANG , Wei YAN , Yimin ZHANG . SHARP WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE HIGHER-ORDER DISPERSIVE EQUATION[J]. Acta mathematica scientia, Series B, 2017 , 37(4) : 1061 -1082 . DOI: 10.1016/S0252-9602(17)30058-9

References

[1] Bejenaru I, Tao T. Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J Funct Anal, 2006, 233:228-259
[2] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part Ⅱ:The KdV equation. Geom Funct Anal, 1993, 3:107-156
[3] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part Ⅱ:The KdV equation. Geom Funct Anal, 1993, 3:209-262
[4] Bourgain J. Periodic Korteweg-de Vries equation with measures as initial data. Selecta Math, 1997, 3:115-159
[5] Constantin P, Saut J C. Local smoothing properties of dispersive equations. J Amer Math Soc, 1988, 1:413-439
[6] Ginibre J, Tsutsumi Y, Velo G. On the Cauchy problem for the Zakharov System. J Funct Anal, 1997, 151:384-436
[7] Guo Z H. Global well-posedness of the Korteweg-de Vries equation in H^-3/4. J Math Pures Appl, 2009, 91:583-597
[8] Guo Z H, Kwak C, Kwon S. Rough solutions of the fifth-order KdV equations. J Funct Anal, 2013, 265:2791-2829
[9] Guo Z H, Wang B X. Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. J Diff Eqns, 2009, 246:3864-3901
[10] Grünrock A. On the hierarchies of higher order mKdV and KdV equations. Central European J Math, 2010, 8:500-536
[11] Ionescu A D, Kenig C E. Global well-posedness of the Benjamin-Ono equation in low-regularity spaces. J Amer Math Soc, 2007, 20:753-798
[12] Ionescu A D, Kenig C E, Tataru D. Global well-posedness of the KP-I initial-value problem in the energy space. Invent Math, 2008, 173:265-304
[13] Kato T. Well-posedness for the fifith order KdV equation. Funkcial Ekvac, 2012, 55:17-53
[14] Kenig C E, Ponce G, Vega L. Higher-order nonlinear dispersive equations. Proc Amer Math Soc, 1994, 122:157-166
[15] Kenig C E, Pilod D. Well-posedness for the fifth-order KdV equation in the energy space. Trans Amer Math Soc, 2015, 367:2551-2612
[16] Kishimoto N. Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity. Diff Int Eqns, 2009, 22:447-464
[17] Kishimoto N, Tsugawa K. Local well-posedness for quadratic nonlinear Schrödinger equations and the "good" Boussinesq equation. Diff Int Eqns, 2010, 23:463-493
[18] Li Y S, Huang J H, Yan W. The Cauchy problem for the Ostrovsky equation with negative dispersion at the critical regularity. J Diff Eqns, 2015, 259:1379-1408
[19] Pilod D. On the Cauchy problem for higher-order nonlinear dispersive equations. J Diff Eqns, 2008, 245:2055-2077
[20] Wang H, Cui S. Well-posedness of the Cauchy problem of a water wave equation with low regularity initial data. Math. Computer Mode, 2007, 45:1118-1132
[21] Yan W, Li Y S, Li S M. Sharp well-posedness and ill-posedness of a higher-ordermodified Camassa-Holm equation. Diff Int Eqns, 2012, 25:1053-1074
[22] Yan W, Li Y S. Ill-posedness of madified Kawahara equation and Kaup-Kupershmidt equation. Acta Math Sci, 2012, 32B:710-716

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