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

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

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (2): 385-394.doi: 10.1016/S0252-9602(17)30009-7

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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).

Abstract

Abstract:

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].

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

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.

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References

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

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