GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

Dexing KONG; Qi LIU

doi:10.1016/S0252-9602(18)30780-X

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 3: 745 - 755

DOI: https://doi.org/10.1016/S0252-9602(18)30780-X

Articles

GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION

Dexing KONG ,
Qi LIU

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1. School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China;
2. Department of Applied Mathematics, College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Received date: 2016-12-12

Revised date: 2017-09-20

Online published: 2018-06-25

Supported by

This work is supported in part by the NNSF of China (11271323, 91330105), the Zhejiang Provincial Natural Science Foundation of China (LZ13A010002), and the Science Foundation in Higher Education of Henan (18A110036).

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Abstract

In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation
(∂²g_ij)/(∂t²) + μ/((1 + t)^λ) (∂g_ij)/∂t=-2R_ij,
on Riemann surface. On the basis of the energy method, for 0 < λ ≤ 1, μ > λ + 1, we show that there exists a global solution g_ij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric g_ij remains uniformly bounded.

Key words： Hyperbolic geometry flow; time-dependent damping; classical solution; energy method; global existence

Cite this article

Dexing KONG , Qi LIU . GLOBAL EXISTENCE OF CLASSICAL SOLUTIONS TO THE HYPERBOLIC GEOMETRY FLOW WITH TIME-DEPENDENT DISSIPATION[J]. Acta mathematica scientia, Series B, 2018 , 38(3) : 745 -755 . DOI: 10.1016/S0252-9602(18)30780-X

References

[1] Kong D X, Dai W R, Liu K F. Dissipative hyperbolic geometry flow. Asian Journal of Mathematics, 2008, 12(3):345-364
[2] Hou F, Witt I, Yin H C. On the global existence and blowup of smooth solutions of 3-D compressible Euler equations with time-depending damping. arXiv:1510.04613
[3] Kong D X. Hyperbolic geometry flow//The proceedings of ICCM 2007. Vol Ⅱ. Beijing:Higher Educational Press, 2007:95-110
[4] Kong D X, Liu K F. Wave character of metric and hyperbolic geometry flow. J Math Phys, 2007, 48(10):103508
[5] Kong D X, Liu K F, Wang Y Z. Life-span of classical solutions to hyperbolic geometry flow in two space variables with decay initial data. Comm Part Diff Eq, 2010, 36(1):162-184
[6] Kong D X, Liu K F, Xu D L. The hyperbolic geometry flow on Riemann surfaces. Comm Part Diff Eq, 2009, 34(4/6):553-580
[7] Kong D X, Liu Q, Song C M. Global existence and asymptotic behavior of classical solutions to the dissipative hyperbolic geometry flow in two space variables (to appear)
[8] Kong D X, Wang J H. Einstein's hyperbolic geometry flow. J Hyperbolic Diff Eq, 2014, 11(2):249-267
[9] Kong D X, Wang Y Z. Long-time behaviour of smooth solutions to the compressible Euler equations with damping in several space variables. IMA J Appl Math, 2012, 77:473-494
[10] Liu F G. Global classical solutions to the dissipative hyperbolic geometric flow on Riemann surfaces (Chinese). Chinese Ann Math Ser A, 2009, 30:717-726
[11] Pan X H. Remarks on 1-D Euler equations with time-depending damping. arXiv:1510.08115v1

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