Acta mathematica scientia, Series B >
ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R2
Received date: 2015-06-06
Revised date: 2016-09-15
Online published: 2017-04-25
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).
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) ∈ Hs(R2)×Hs-1(R2). It is shown that the IVP is global well-posedness in Hs(R2)×Hs-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].
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 . DOI: 10.1016/S0252-9602(17)30009-7
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