ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R2

doi:10.1016/S0252-9602(17)30009-7

数学物理学报(英文版) ›› 2017, Vol. 37 ›› Issue (2): 385-394.doi: 10.1016/S0252-9602(17)30009-7

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²

张再云^1,2, 黄建华², 孙明保¹

1. School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China;
2. College of Science, National University of Defense Technology, Changsha 410073, China

收稿日期:2015-06-06 修回日期:2016-09-15 出版日期:2017-04-25 发布日期:2017-04-25
作者简介:Zaiyun ZHANG,E-mail:zhangzaiyun1226@126.com;Jianhua HUANG,E-mail:jhhuang32@nudt.edu.cn;Mingbao SUN,E-mail:sun mingbao@163.com
基金资助:
This work was supported by Hunan Provincial Natural Science Foundation of China (2016JJ2061), Scientific Research Fund of Hunan Provincial Education Department (15B102), China Postdoctoral Science Foundation (2013M532169, 2014T70991), NNSF of China (11671101, 11371367, 11271118), the Construct Program of the Key Discipline in Hunan Province (201176), and the aid program for Science and Technology Innovative Research Team in Higher Education Institutions of Hunan Province (2014207).

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²

Zaiyun ZHANG^1,2, Jianhua HUANG², Mingbao SUN¹

1. School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China;
2. College of Science, National University of Defense Technology, Changsha 410073, China

Received:2015-06-06 Revised:2016-09-15 Online:2017-04-25 Published:2017-04-25
Contact: Zaiyun ZHANG,E-mail:zhangzaiyun1226@126.com E-mail:zhangzaiyun1226@126.com
Supported by:
This work was supported by Hunan Provincial Natural Science Foundation of China (2016JJ2061), Scientific Research Fund of Hunan Provincial Education Department (15B102), China Postdoctoral Science Foundation (2013M532169, 2014T70991), NNSF of China (11671101, 11371367, 11271118), the Construct Program of the Key Discipline in Hunan Province (201176), and the aid program for Science and Technology Innovative Research Team in Higher Education Institutions of Hunan Province (2014207).

摘要/Abstract

摘要：

In this article, we investigate the initial value problem(IVP) associated with the defocusing nonlinear wave equation on R² as follows:

where the initial data (u₀, u₁) ∈ H^s(R²)×H^s-1(R²). It is shown that the IVP is global well-posedness in H^s(R²)×H^s-1(R²) for any 1 > s > 2/5. The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy[1].

关键词: Defocusing nonlinear wave equation, global well-posedness, I-method, linear-nonlinear decomposition, below energy space

Abstract:

Key words: Defocusing nonlinear wave equation, global well-posedness, I-method, linear-nonlinear decomposition, below energy space

张再云, 黄建华, 孙明保. ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²[J]. 数学物理学报(英文版), 2017, 37(2): 385-394.

Zaiyun ZHANG, Jianhua HUANG, Mingbao SUN. ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²[J]. Acta mathematica scientia,Series B, 2017, 37(2): 385-394.

参考文献

[1] Roy T. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on R³. Disc Cont Dyn Sys A, 2008, 24(4):1307-1323
[2] Lindbald H, Sogge C. On existence and scattering with minimal regularity for semilinear wave equations. J Funct Anal, 1995, 130:357-426
[3] Kenig C, Ponce G, Vega L. Global well-posedness for semi-linear wave equations. Comm Partial Differ Equ, 2000, 25:1741-1752
[4] Bourgain J. Refinement of Strichartz inequality and applications to 2D-NLS with critical nonlinearity. Internat Math Res Notices, 1998, 5:253-283
[5] Gallagher I, Planchon F. On global solutions to a dofocusing semi-linear wave equation. Revista Math Iberoamericana, 2003, 19:161-177
[6] Bahouri H, Chemin J. On global well-posedness for defocusing cubic wave equation. Internat Math Res Notices, 2006, Article ID 54873:1-12
[7] Klainerman S, Tataru D. On the optimal local regularity for the Yang-Mills equations in R⁴⁺¹. J Amer Math Soc, 1999, 12:93-116
[8] Roy T. Global well-posedness for the radial defocusing cubic wave equation on R³ and for rough data. EJDE, 2007, 2007:1-22
[9] Fonseca G E. Global well-posedness fro two dimensional semilinear wave equations. Revista Colombiana de Mathemáticas, 2006, 34:91-101
[10] Bourgain J. New global well-posedness results for non-linear Schrödinger equations. AMS, 1999
[11] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. Sharp global well-posedness for KdV and modified Kdv on R and T. J Amer Math Soc, 2003, 16:705-749
[12] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. A refined global well-posedness result for Schrödinger equations with derivatives. SIAM J Math Anal, 2002, 34:64-86
[13] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. Global well-posedness and scattering for rough solutions of a nonlinear Schrödinger equation on R³. Commmunication on Pure and Applied Math, 2004, 57(8):987-1014
[14] Ginebre J, Velo G. Generalized Strichartz inequalities for the wave equation. J Func Anal, 1995, 133:50-68
[15] Keel M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120:955-980
[16] Tao T. Nonlinear dispersive equations, volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences. Local and global analysis. Washington, DC, 2006
[17] Roy T. Global analysis of the defocusing cubic wave equation in dimension 3(preprint). 2008
[18] Wang B X, Huo Z H, Chao C C, Guo Z H. Harmonic analysis method for nonlinear evolution equations. World Scientific Press, 2011
[19] Coifman R R, Meyer Y. Commutateurs d'integrales singulières et opérateurs multilinéaires. Ann Inst Fourier (Grenoble), 1978, 28:177-202
[20] Guo Z H, Wang B X. Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. J Differential Equations, 2009, 246:3864-3901
[21] Li Y S, Wu Y H, Xu G X. Low regularity global solutions for the focusing mass-critical nonlinear Schrodinger equation in R. SIAM J Math Anal, 2011, 43:322-340
[22] Li Y S, Wu Y H, Xu G X. Global well-posedness for periodic mass-critical nonlinear Schrodinger equation. J Differential Equations, 2011, 250:2715-2736
[23] Yang X Y, Li Y S. Global well-posedness for a fifth-order shallow water equation in Sobolev spaces. J Differential Equations, 2010, 248:1458-1472

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R²

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