强阻尼波动方程吸引子的正则性及其逼近

Abstract

Abstract:

In this paper the authors consider the initial boundary value problem of the strongly damped wave equation. By using the analytic property of the semigroup generated by the principal operator of the equation in the phase space, they show the smoothing effect of the solution, which is much different from the case of weakly dissipative wave equations. Thus they obtain the regularity of the global attractor, and construct inertial manifolds of the equation, just as usually done in the selfadjoint cases.

Key words: Ｓｔｒｏｎｇｌｙｄａｍｐｅｄｗａｖｅｅｑｕａｔｉｏｎ, Ａｎａｌｙｔｉｃｓｅｍｉｇｒｏｕｐ, Ｒｅｇｕｌａｒｉｔｙ, Ｇｌｏｂａｌａｔｔｒａｃｔｏｒ, Ａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄ

CLC Number:

３５Ｌ２０，３５Ｂ４０，５８Ｆ３９

Li Yongsheng Zhang Weiguo. Regularity and Approximation of the Attractor for the Strongly Damped Wave Equation[J].Acta mathematica scientia,Series A, 2000, 20(3): 342-350.

Trendmd

References

１　ＢａｂｉｎＡＶ，ＶｉｓｈｉｋＭＩ．Ｒｅｇｕｌａｒａｔｔｒａｃｔｏｒｓｏｆｓｅｍｉｇｒｏｕｐｓａｎｄｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＰｕｒｅｓｅｔＡｐｐｌ，１９８３，６２（４）：４４１－４４９
２　ＢａｂｉｎＡＶ，ＶｉｓｈｉｋＭＩ．Ａｔｔｒａｃｔｏｒｓｏｆｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＮｏｒｔｈＨｏｌｌａｎｄ：Ａｍｓｔｅｒｄａｍ，１９９２
３　ＢｅｒｋａｌｉｅｖＺＢ．Ａｎａｔｔｒａｃｔｏｒｏｆｓｏｍｅｑｕａｓｉｌｉｎｅａｒｓｙｓｔｅｍｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｖｉｓｃｏｅｌａｓｔｉｃｔｅｒｍｓ．ＲｕｓｓｉａｎＭａｔｈＳｕｒｖｅｙ，４０（１）：２０９－２１０
４　ＢｅｒｋａｌｉｅｖＺＢ．Ａｎａｔｔｒａｃｔｏｒｏｆａｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｏｆｖｉｓｃｏｅｌａｓｔｉｃｉｔｙ．ＭｏｓｃｏｗＵｎｉｖＭａｔｈＢｕｌｌ，１９８５，４０（５）：６１－６３
５　ＣｈａｎｇＱＳ，ＧｕｏＢＬ．ＦｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅｓｏｌｕｔｉｏｎｆｏｒｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎU_U－U_XX＝（σ（U_X））_X＋U_xxt－f（u）．ＪＣｏｍｐｔＭａｔｈ，１９８４，１２（４）：２９７－３０４
６　ＣｈｏｗＳＮ，ＬｕＫ．ＩｎｖａｒｉａｎｔｍａｎｉｆｏｌｄｓｆｏｒｆｌｏｗｓｉｎＢａｎａｃｈｓｐａｃｅｓ．ＪＤｉｆｆＥｑｓ，１９８８，７４：２８５－３１７
７　ＣｈｕｅｓｈｏｖＩＤ．Ｏｎａｃｏｎｓｔｒｕｃｔｉｏｎｏｆａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓｆｏｒｓｅｃｏｎｄｏｒｄｅｒｉｎｔｉｍｅｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＮｏｎｌｉｎｅａｒＡｎａｌ，１９９６，２６（５）：１００７－１０２１
８　ＤｅｂｕｓｓｃｈｅＡ，ＭａｒｉｏｎＭ．Ｏｎｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｆａｍｉｌｉｅｓｏｆａｐｐｒｏｘｉｍａｔｅｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓ．ＪＤｉｆｆＥｑｓ，１９９２，１００：１７３－２０１
９　ＦｉｔｚｇｉｂｂｏｎＷＥ．Ｓｔｒｏｎｇｌｙｄａｍｐｅｄｑｕａｓｉｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＡｎａｌＡｐｐｌ，１９８１，７９：５３６－５５０
１０　ＧｈｉｄａｇｌｉａＪＭ，ＴｅｍａｍＲ．Ａｔｔｒａｃｔｏｒｓｆｏｒｄａｍｐｅｄｎｏｎｌｉｎｅａｒｈｙｐｅｒｂｏｌｉｃｅｑｕａｔｉｏｎｓ．ＪＭａｔｈＰｕｒｅｓｅｔＡｐｐｌ，１９８７，６６：２７３－３１９
１１　ＧｈｉｄａｇｌｉａＪＭ，ＴｅｍａｍＲ．ＤｉｍｅｎｓｉｏｎｏｆｔｈｅｕｎｉｖｅｒｓａｌａｔｔｒａｃｔｏｒｄｅｓｃｒｉｂｉｎｇｔｈｅｐｅｒｉｏｄｉｃａｌｌｙｄｒｉｖｅｎｓｉｎｅＧｏｒｄｏｎｅｑｕａｔｉｏｎｓ，ＴｒａｎｓｐｏｒｔＴｈｅｏｒｙＳｔａｔｉｓｔ．Ｐｈｙｓ，１９８７，１６（２～３）：２５３－２６５
１２　ＣｈｅｎＦＸ，ＧｕｏＢＬ，ＺｈａｎｇＰ．Ｌｏｎｇｔｉｍｅｂｅｈａｖｏｉｒｏｆｓｔｒｏｎｇｌｙｄａｍｐｅｄｗａｖｅｅｑｕａｔｉｏｎｓ．ＪＤｉｆｆＥｑｕａｔｉｏｎｓ，１９９８，１４７：２３１－２４１
１３　ＬｏｍｄａｈｌＰＳ，ＳｏｅｒｅｎｓｅｎＯＨ，ＣｈｒｉｓｔｉａｎｓｅｎＰＬ．ＳｏｌｉｔｏｎｅｘｃｉｔａｔｉｏｎｓｉｎＪｏｓｅｐｈｓｏｎｔｕｎｎｅｌｊｕｎｃｔｉｏｎｓ．ＰｈｙｓＲｅｖＢ，１９８２，２５：５７３７－５７４８
１４　ＭａｓｓａｔＰ．Ｌｉｍｉｔｉｎｇｂｅｈａｖｉｏｒｆｏｒｓｔｒｏｎｇｌｙｄａｍｐｅｄｎｏｎｌｉｎｅａｒｗａｖｅｅｑａｔｉｏｎｓ．ＪＤｉｆｆＥｑｓ，１９８３，４８：３３４－３４９
１５　ＮａｇｕｍｏＪ，ＡｒｉｍｏｔｏＪ，ＹｏｓｈｉｚａｗａＳ．Ａｎａｃｔｉｖｅｐｕｌｓｅｔｒａｎｓｍｉｓｓｉｏｎｌｉｎｅｓｉｍｕｌａｔｉｎｇｎｅｒｖｅａｘｏｎ．ＰｒｏｃＩＲＥ，１９６２，５０：２０６１－２０７０
１６　ＰａｏＣＶ．Ａｍｉｘｅｄｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｒｉｓｉｎｇｉｎｎｅｕｒｏｐｈｙｓｉｌｏｇｙ．ＪＭａｔｈＡｎａｌＡｐｐｌ，１９７５，５２：１０５－１１９
１７　ＰａｚｙＡ．Ｓｅｍｉｇｒｏｕｐｓｏｆｌｉｎｅａｒｏｐｅｒａｔｏｒｓａｎｄａｐｐｌｉｃａｔｉｏｎｓｔｏｐａｒｔｉａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ．Ｂｅｒｌｉｎ：Ｓｐｒｉｎｇｅｒ，１９８３
１８　ＳｅｌｌＧＲ，ＹｏｕＹ．Ｉｎｅｒｔｉａｌｍａｎｉｆｏｌｄｓ：Ｔｈｅｎｏｎｓｅｌｆａｄｊｏｉｎｔｃａｓｅ．ＪＤｉｆｆＥｑｓ，１９９２，９６：２０３－２５５
１９　ＳｈｏｗａｌｔｅｒＲＥ．Ｒｅｇｕｌａｒｉｚａｔｉｏｎａｎｄａｐｐｒｏｘｉｍａｔｉｏｎｏｆｓｅｃｏｎｄｏｒｄｅｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓ．ＳＩＡＭＪＭａｔｈＡｎａｌ，１９７６，７：４６１－４７２
２０　ＴｅｍａｍＲ．Ｉｎｆｉｎｉｔｅｄｉｍｅｎｓｉｏｎａｌｄｙｎａｍｉｃａｌｓｙｓｔｅｍｓｉｎｍｅｃｈａｎｉｃｓａｎｄｐｈｙｓｉｃｓ．Ｂｅｒｌｉｎ：Ｓｐｒｉｎｇｅｒ，１９８８
２１　ＷｅｂｂＧＦ．Ｅｘｉｓｔｅｎｃｅａｎｄａｓｙｍｐｔｏｔｉｃｂｅｈａｖｉｏｒｆｏｒａｓｔｒｏｎｇｌｙｄａｍｐｅｄｎｏｎｌｉｎｅａｒｗａｖｅｅｑｕａｔｉｏｎ．ＣａｎａｄＪＭａｔｈ，１９８０，３２：６３１－６４３
２２　ＺｈａｎｇＷＧ．ＴｈｅＵｎｉｖｅｒｓａｌｃｏｍｐａｃｔａｔｔｒａｃｔｏｒｆｏｒｍｉｘｅｄｅｑｕａｔｉｏｎｓｏｆｎｏｎｌｉｎｅａｒｗａｖｅａｎｄｎｅｒｖｅｃｏｎｄｕｃｔ．ＡｃｔａＭａｔｈＡｐｐｌＳｉｎｉｃａ，１９９８，２１（３）：３３９－３５２

[1]	Zhao Jihong. Logarithmical Regularity Criteria in Terms of Pressure for the Three Dimensional Dissipative System Modeling Electro-Hydrodynamics [J]. Acta mathematica scientia,Series A, 2018, 38(3): 549-564.
[2]	Li Qingwei, Gao Wenjie, Han Yuzhu. Existence of Solutions to Some Singular Elliptic Problems with Degenerate Coercivity [J]. Acta mathematica scientia,Series A, 2017, 37(5): 877-894.
[3]	Gao Hongya, Jia Miaomiao. Local Regularity and Local Boundedness for Solutions to Obstacle Problems [J]. Acta mathematica scientia,Series A, 2017, 37(4): 706-713.
[4]	Ba Na, Zheng Lie. Partial Schauder Estimates for the Heat Equation [J]. Acta mathematica scientia,Series A, 2017, 37(2): 307-312.
[5]	Zhang Lina, Li Junping, Geng Shifeng. The Stability Property of Generalized Nonlinear Markov Branching Models with Resurrection [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1190-1206.
[6]	ZHANG Zu-Jin. An Improved Regularity Criterion for the 3D Navier-Stokes Equations in Terms of Two Entries of the Velocity Gradient [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1327-1335.
[7]	Leszek GASI′ NSKI, Nikolaos S. PAPAGEORGIOU. POSITIVE SOLUTIONS FOR PARAMETRIC EQUIDIFFUSIVE p-LAPLACIAN EQUATIONS [J]. Acta mathematica scientia,Series A, 2014, 34(3): 610-618.
[8]	MA Wen-Jun, MA Qiao-Zhen, GAO Pei-Ming. Regularity and Global Attractor of Nonlinear Elastic Rod Oscillation Equation [J]. Acta mathematica scientia,Series A, 2014, 34(2): 445-453.
[9]	ZHOU Shu-Qing, HU Zhen-Hua, PENG Dong-Yun. Global Regularity for Very Weak Solutions to Obstacle Promlems Corresponding to a Class of A-Harmonic Equations [J]. Acta mathematica scientia,Series A, 2014, 34(1): 27-38.
[10]	JIN Shou-Bo, CHEN Pan-Feng, SUN Shan-Hui. Regularity of the Solution to the Dissipative Linear Boltzmann Equation [J]. Acta mathematica scientia,Series A, 2011, 31(6): 1662-1668.
[11]	GAO Hong-Ya, HAN Xiao-Fan. Regularity Results for Div-Curl Fields and Applications [J]. Acta mathematica scientia,Series A, 2011, 31(2): 378-386.
[12]	SUN Jian-Zhu, FAN Ji-Shan. Regularity Criteria for a Two-fluid Model of the Truncated Euler Equations [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1693-1698.
[13]	WU Hong-Wei. Regularity Criteria for the Compressible Magneto-hydrodynamic Equations [J]. Acta mathematica scientia,Series A, 2010, 30(3): 593-602.
[14]	LIU Xiao-Feng. New Regularity Conditions for Magneto-hydrodynamics Equations [J]. Acta mathematica scientia,Series A, 2010, 30(2): 335-343.
[15]	GAO Hong-Y, HE Qian, NIU Hong-Ling, CHU Yu-Ming. Local Regularity for Very Weak Solutions of Obstacle Problems to the A-Harmonic Operators [J]. Acta mathematica scientia,Series A, 2009, 29(5): 1291-1297.

Regularity and Approximation of the Attractor for the Strongly Damped Wave Equation

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10