一类粘性色散波方程的全局吸引子
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摘要
非线性现象是自然界中普遍存在的一种重要现象。最近几十年来,物理、力学、化学、生物、工程、航空航天、医学、经济和金融等领域中诞生了许多非线性偏微分方程,但是由于方程的非线性以及本身的复杂性,使得对这些方程的研究具有很大的挑战性。本文研究了两类有着深刻物理背景的非线性偏微分方程,即粘性Fomberg-Whitham方程和一类粘性色散波方程。本文研究的主要内容:引进整体吸引子的概念,考虑在周期边界条件下F0nlberg-Whitham方程和一类粘性色散波方程的全局解和全局吸引子存在性问题。
第三章研究带粘性项的Fornberg-Whitham方程,运用伽辽金方法得到了L~2(R)空间下全局解的存在性,结果表明了在L~2(R)空间中F0rnberg-Whitham方程存在唯一的全局解。接着利用Sobolev插值不等式以及利用关于时间t的先验估计等方法证明了该方程在H~2(R)空间上吸收集的存在性,最后通过证明方程的解半群S(t)是一个紧算子得到Fornberg-Whitham方程全局吸引子的存在性。
第四章研究了一类粘性色散波方程。这一章用了和第三章一样的讨论方法。首先得到了全局解的存在性,接着讨论了方程解半群吸收集在H~2(R)空间上的存在性,最后证明了该方程存在全局吸引子。
Nonlinearity is universal and important phenomenon in nature. In recent years, many nonlinear partial differential equations were derived from physics, mechanics, chemistry, biology, engineering, aeronautics, medicine, economy, finance and many other fields. Because of the non-linearity and complexity of themselves, it is a big challenge to deal with them. In the paper, we study two nonlinear partial differential dispersive equations, that is, viscous Fornberg-Whitham equation and a class of viscous nonlinear dispersive wave equations. In this paper, we introduce the concept of global attractor and get the existence of global solution and the global attractor of a viscosity Fornberg-Whitham equation and the viscous nonlinear dispersive wave equation on periodical boundary condition.
In the third chapter, the Galerkin Procedure is applied to show the existence of the global solution of Fornberg-Whitham equation in L~2(R). The Sobolev interpolation inequality and prior estimate on time-t are applied to show the existence of attracting set. Moreover, we prove the semi-group of the solution operator is a compact operator. Finally, we get the existence of the global attractor of a viscous Fornberg-Whitham equation in H~2(R).
In the forth chapter, we give a study on a class of viscous nonlinear dispersive wave equations. We use the same discussion method as the third chapter. First we show the existence of global solution of the viscous nonlinear dispersive wave equation, then the existence of attracting set is obtained in H~2(R). Finally we proof that the equation has a global attractor.
第三章研究带粘性项的Fornberg-Whitham方程,运用伽辽金方法得到了L~2(R)空间下全局解的存在性,结果表明了在L~2(R)空间中F0rnberg-Whitham方程存在唯一的全局解。接着利用Sobolev插值不等式以及利用关于时间t的先验估计等方法证明了该方程在H~2(R)空间上吸收集的存在性,最后通过证明方程的解半群S(t)是一个紧算子得到Fornberg-Whitham方程全局吸引子的存在性。
第四章研究了一类粘性色散波方程。这一章用了和第三章一样的讨论方法。首先得到了全局解的存在性,接着讨论了方程解半群吸收集在H~2(R)空间上的存在性,最后证明了该方程存在全局吸引子。
Nonlinearity is universal and important phenomenon in nature. In recent years, many nonlinear partial differential equations were derived from physics, mechanics, chemistry, biology, engineering, aeronautics, medicine, economy, finance and many other fields. Because of the non-linearity and complexity of themselves, it is a big challenge to deal with them. In the paper, we study two nonlinear partial differential dispersive equations, that is, viscous Fornberg-Whitham equation and a class of viscous nonlinear dispersive wave equations. In this paper, we introduce the concept of global attractor and get the existence of global solution and the global attractor of a viscosity Fornberg-Whitham equation and the viscous nonlinear dispersive wave equation on periodical boundary condition.
In the third chapter, the Galerkin Procedure is applied to show the existence of the global solution of Fornberg-Whitham equation in L~2(R). The Sobolev interpolation inequality and prior estimate on time-t are applied to show the existence of attracting set. Moreover, we prove the semi-group of the solution operator is a compact operator. Finally, we get the existence of the global attractor of a viscous Fornberg-Whitham equation in H~2(R).
In the forth chapter, we give a study on a class of viscous nonlinear dispersive wave equations. We use the same discussion method as the third chapter. First we show the existence of global solution of the viscous nonlinear dispersive wave equation, then the existence of attracting set is obtained in H~2(R). Finally we proof that the equation has a global attractor.
引文
[1]R.Temam.Infinite-Dimensional Dynamical Systems in Mechanics and Phsics.Apple.Math.Sci.68,Springer-Vedag.New York.1988
[2]J.K.Hale.Asymptotic behavior of dissipative systems[M].Providence.RI:AMS.1983:35-60
[3]I.D.Chueshov.Global attractors for nonlinear problems of mathematical phsics[J].Russian Math.1993(48):135-162
[4]Ghidaglia J M.Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time.J.Diff.Eqs.1988(74):369-390
[5]Ghidaglia J M.A Note on strong convergence towards attractors of damped forced KdV equations.J.Diff.Eqs.1994(110):356-359
[6]郭柏灵.非线性演化方程.上海:上海科技教育出版社,1995
[7]LixinTian,Zhenyuan Xu.The research of longtime dynamic behavior in weakly damped forced KdV equation.Applied Mathematics and Mechanics.18(10)(101997):953-958
[8]Moise I,Rosa R.On the regularity of the global attractor of a weakly damped forced Korteweg-de Vries equation.Adv.Diff.Eqs.1997(2):257-256
[9]Laurencot P.Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line.Czechoslovak Math.J.1998(48):85-94
[10]LixinTian,Danping Ding.The research of blow-up in 2D weakly damped KdV equation on thin domain.Journal of Jiangsu University.2000(21):106-110
[11]Lixin Tian,Yurong Liu,Zengrong Liu.Local attractors for the weakly damped forced KdV equation in thin 2D domains.Applied Mathematics and Mechanics.2000(21):1021-1027
[12]Lixin Tian,Zhenyuan Xu,Zengrong Liu.Attractors in dissipative soliton equation.Appli.Math.Mech.1994(15):571-579
[13]LixinTian,Danping Ding.The research of blow-up in 2D weakly damped KdV equation on thin domain.Journal of Jiangsu University.2000(21):106-110
[14]Olivier Goubet,Ricardo M S Rosa.Asymptotic smoothing and the global attractor of weakly damped forced KdV equations on the real line.J.Diff.Eqs.2002(185):25-53
[15]Wenbin Zhang,Lixin Tian,Dejun Peng.Asymptotic smoothing and the global attractor of weakly damped forced KdV equations on the real line.Journal of Jiangsu University of Science and Technology.2005(12):38-42
[16]Danping Ding,Lixin Tian.The attractor in dissipative Camassa-Holm equation.ActaMath.Appl.Sin.2004(27):536-545
[17]Danping Ding,Lixin Tian.The study on solution of dissipative Camassa-Holm equation with weak dissipation.Commun.Pure Appl.Anal.2006(5):483-493
[18] Suping Qian. Nonclassical symmetry reductions for coupled KdV equation. International Journal of Nonlinear Science. 2006 (2): 97-103
[19] Danping Ding, Lixin Tian. The study of solution of dissipative Camassa-Holm equation on total space. International Journal of Nonlinear Science. 2006 (1): 37-42
[20] Lixin Tian, Jinling Fan, Ruihua Tian. The attractor on viscosity peakon b-family of equations. International Journal of Nonlinear Science. 2007 (3): 163-170
[21] Lixin Tian, Jinling Fan. The attractor on viscosity Degasperis-Procesi equation. Nonlinear Anal: Real Word Appl. 2007 (9):1461-1473
[22] Lixin Tian, Ruihua Tian, Jinling Fan. The global attractor for the viscous weakly damped forced Krteweg-de Vries equations in H~1(R) . International Journal of Nonlinear Science. 2008 (1): 3-10
[23] Wenxia Chen, Lixin Tian, Xiaoyan Deng. The global attractor and numerical simulation of a forced weakly damped MKdV equation. Nonlinear Anal: Real Word Appl. 2007:1468-1218
[24] Anna Gao, Lixin Tian. The global attractor of the viscous damped forced Ostrovsky equation. Nonlinear Anal: Real World Appl. 2009(10):2894-2904
[25] A. Degasperis, M. Procesi. Asymptotic integrability. In: Symmetry and Perturbation Theory. Singapore: World Scientific 1999: 23-37
[26] R. Camassa, D.D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993 (71): 1661-1664
[27] R.S. Johnson. Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 2002 (455): 63-82
[28] A. Constantin, D. Lannes. The hydro dynamical relevance of the Camassa -Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 2009 (192): 165-186
[29] J. Lenells. Traveling wave solutions of the Camassa-Holm equation. J.Differential Equations 2005 (217): 393-430
[30] A. Constantin, W. Strauss. Stability of peakons. Comm. Pure Appl. Math. 2000 (53): 603-610
[31] A. Constantin, W. Strauss. Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 2002 (12): 415-422
[32] R. Beals, D. Sattinger, J. Szmigiel-ski. Multi-peakons and a theorem of Stieltjes. Inverse Problems 1999 (15): L1-L4
[33] R. S. Johnson. On solutions of the Camassa-Holm equation. Proc. Roy. Soc. London A 2003 (459): 1687-1708
[34] A. Bressan, A. Constantin. Global conservative solutions of the Camassa -Holm equation. Arch. Ration. Mech. Anal. 2007 (183): 215-239
[35] A. Bressan, A. Constantin. Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 2007 (5): 1-27
[36] Jiangbo Zhou, Lixin Tian. Blow-up of solution of an initial boundary value problem for a generalized Camassa-Holm equation. Phy. Lett A. 2008 (372): 3659-3666
[37] A. Degasperis, D.D. Holm, A.N.W. Hone. A new integral equation with peakon solutions. Theo. Math. Phys. 2002 (133): 1463-1474
[38] H.R. Dullin, GA. Gottwald, D.D. Holm. Camassa-Holm, Korteweg-de Veris and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 2003 (33): 73-79
[39] J. Lenells. Traveling wave solutions of the Degasperis-Procesi equation. J. Math. Anal. Appl. 2005 (306): 72-82
[40] Lixin Tian, Guochang Fang, Guilong Gui. Well-posedness problem and blow-up for an integrable shallow water equation with strong dispersive term. International Journal of Nonlinear Science 2006 (1): 3-12
[41] GM. Coclite, K.H. Karlsen, N.H. Risebro. Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, Pure Math. 2006 (6): 1-23
[42] H. Lundmark, Formation. Dynamics of shock waves in the Degasperis -Procesi equation. Journal of Nonlinear Science 2007 (17): 169-198
[43] Lixin Tian, Xiuming Li. On the well-posedness problem for the generalized Degasperis- Procesi equation. International Journal of Nonlinear Science. 2006 (2): 67-76
[44] Liqin Yu, Lixin Tian, Xuedi Wang. The bifucation and peakon for Degasperis-Procesi equation. Chaos, Soliton & Fractals 2006 (30): 956-966
[45] Lixin Tian, Chunyu Shen. Optimal control of the viscous Degasperis-Procesi equation. J. Math. Phy. 2007 (48): 13-16
[46] Lixin Tian, Xiuming Li. Well-posedness for a New Completely Integrable Shallow Water Wave Equation. International Journal of Nonlinear Science. 2007 (4): 83-91
[47] Lixin Tian, Meijie Ni. Blow-up phenomena for the periodic Degasperis -Procesi equation with strong dispersive term. International Journal of Nonlinear Science. 2006 (2): 177-182
[48] Lixin Tian, Meijie Ni. Blow-up phenomena for the Degasperis-Procesi equation with strong dispersive term. International Journal of Nonlinear Science. 2007 (3): 187-194
[49] Jiangbo Zhou, Anchan Mai. A Mixed Initial Boundary Value Problem for the Degasperis-Procesi Equation on a Bounded Interval. International Journal of Nonlinear Science. 2008 (5): 255-260
[50] Jiangbo Zhou. Global Existence of Solution to an Initial Boundary Value Problem for the Degasperis-Procesi Equation. International Journal of Nonlinear Science. 2007 (4): 141-146
[51] G. B. Whitham. Varantional methods and applications to water wave. Proc. R. Soc. Lond. A 1967 (299): 6-25
[52] B. Fornberg, G. B. Whitham. A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A1978 (289): 373-404
[53] R. Ivanov. On the integrability of a class of nonlinear dispersive wave equations. J. Nonlinear Math. Phys. 2005 (1294): 462-468
[54] A. Constantin, J. Escher. Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica 1998 (181): 229-243
[55] A. Constantin. The trajectories of particles in Stokes waves. Invent. Math. 2006 (166): 523-535
[56] J. F. Toland. Stokes waves. Topol. Methods Nonlinear Anal. 1996 (7): 1-48
[57] A. Constantin, J. Escher. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 2007 (44): 423-431
[58] Jiangbo Zhou, Lixin Tian. A type of bounded traveling wave solutions for the Fornberg -Whitham equation. J. Math. Anal. Appl. 2008 (346): 255-261
[59] Jiangbo Zhou, Lixin Tian. Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation. Nonlinear Anal: Real World Appl.doi:10. 1016/j.nonrwa.2008.11.014
[60] Babin A V, Vishik M I. Attractor of Evolution Equations. Studies in Math.and its Appl, North-Holland, Amsterdam, London, New York, Tokyo: 1992,25
[2]J.K.Hale.Asymptotic behavior of dissipative systems[M].Providence.RI:AMS.1983:35-60
[3]I.D.Chueshov.Global attractors for nonlinear problems of mathematical phsics[J].Russian Math.1993(48):135-162
[4]Ghidaglia J M.Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time.J.Diff.Eqs.1988(74):369-390
[5]Ghidaglia J M.A Note on strong convergence towards attractors of damped forced KdV equations.J.Diff.Eqs.1994(110):356-359
[6]郭柏灵.非线性演化方程.上海:上海科技教育出版社,1995
[7]LixinTian,Zhenyuan Xu.The research of longtime dynamic behavior in weakly damped forced KdV equation.Applied Mathematics and Mechanics.18(10)(101997):953-958
[8]Moise I,Rosa R.On the regularity of the global attractor of a weakly damped forced Korteweg-de Vries equation.Adv.Diff.Eqs.1997(2):257-256
[9]Laurencot P.Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line.Czechoslovak Math.J.1998(48):85-94
[10]LixinTian,Danping Ding.The research of blow-up in 2D weakly damped KdV equation on thin domain.Journal of Jiangsu University.2000(21):106-110
[11]Lixin Tian,Yurong Liu,Zengrong Liu.Local attractors for the weakly damped forced KdV equation in thin 2D domains.Applied Mathematics and Mechanics.2000(21):1021-1027
[12]Lixin Tian,Zhenyuan Xu,Zengrong Liu.Attractors in dissipative soliton equation.Appli.Math.Mech.1994(15):571-579
[13]LixinTian,Danping Ding.The research of blow-up in 2D weakly damped KdV equation on thin domain.Journal of Jiangsu University.2000(21):106-110
[14]Olivier Goubet,Ricardo M S Rosa.Asymptotic smoothing and the global attractor of weakly damped forced KdV equations on the real line.J.Diff.Eqs.2002(185):25-53
[15]Wenbin Zhang,Lixin Tian,Dejun Peng.Asymptotic smoothing and the global attractor of weakly damped forced KdV equations on the real line.Journal of Jiangsu University of Science and Technology.2005(12):38-42
[16]Danping Ding,Lixin Tian.The attractor in dissipative Camassa-Holm equation.ActaMath.Appl.Sin.2004(27):536-545
[17]Danping Ding,Lixin Tian.The study on solution of dissipative Camassa-Holm equation with weak dissipation.Commun.Pure Appl.Anal.2006(5):483-493
[18] Suping Qian. Nonclassical symmetry reductions for coupled KdV equation. International Journal of Nonlinear Science. 2006 (2): 97-103
[19] Danping Ding, Lixin Tian. The study of solution of dissipative Camassa-Holm equation on total space. International Journal of Nonlinear Science. 2006 (1): 37-42
[20] Lixin Tian, Jinling Fan, Ruihua Tian. The attractor on viscosity peakon b-family of equations. International Journal of Nonlinear Science. 2007 (3): 163-170
[21] Lixin Tian, Jinling Fan. The attractor on viscosity Degasperis-Procesi equation. Nonlinear Anal: Real Word Appl. 2007 (9):1461-1473
[22] Lixin Tian, Ruihua Tian, Jinling Fan. The global attractor for the viscous weakly damped forced Krteweg-de Vries equations in H~1(R) . International Journal of Nonlinear Science. 2008 (1): 3-10
[23] Wenxia Chen, Lixin Tian, Xiaoyan Deng. The global attractor and numerical simulation of a forced weakly damped MKdV equation. Nonlinear Anal: Real Word Appl. 2007:1468-1218
[24] Anna Gao, Lixin Tian. The global attractor of the viscous damped forced Ostrovsky equation. Nonlinear Anal: Real World Appl. 2009(10):2894-2904
[25] A. Degasperis, M. Procesi. Asymptotic integrability. In: Symmetry and Perturbation Theory. Singapore: World Scientific 1999: 23-37
[26] R. Camassa, D.D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993 (71): 1661-1664
[27] R.S. Johnson. Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 2002 (455): 63-82
[28] A. Constantin, D. Lannes. The hydro dynamical relevance of the Camassa -Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 2009 (192): 165-186
[29] J. Lenells. Traveling wave solutions of the Camassa-Holm equation. J.Differential Equations 2005 (217): 393-430
[30] A. Constantin, W. Strauss. Stability of peakons. Comm. Pure Appl. Math. 2000 (53): 603-610
[31] A. Constantin, W. Strauss. Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 2002 (12): 415-422
[32] R. Beals, D. Sattinger, J. Szmigiel-ski. Multi-peakons and a theorem of Stieltjes. Inverse Problems 1999 (15): L1-L4
[33] R. S. Johnson. On solutions of the Camassa-Holm equation. Proc. Roy. Soc. London A 2003 (459): 1687-1708
[34] A. Bressan, A. Constantin. Global conservative solutions of the Camassa -Holm equation. Arch. Ration. Mech. Anal. 2007 (183): 215-239
[35] A. Bressan, A. Constantin. Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. 2007 (5): 1-27
[36] Jiangbo Zhou, Lixin Tian. Blow-up of solution of an initial boundary value problem for a generalized Camassa-Holm equation. Phy. Lett A. 2008 (372): 3659-3666
[37] A. Degasperis, D.D. Holm, A.N.W. Hone. A new integral equation with peakon solutions. Theo. Math. Phys. 2002 (133): 1463-1474
[38] H.R. Dullin, GA. Gottwald, D.D. Holm. Camassa-Holm, Korteweg-de Veris and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 2003 (33): 73-79
[39] J. Lenells. Traveling wave solutions of the Degasperis-Procesi equation. J. Math. Anal. Appl. 2005 (306): 72-82
[40] Lixin Tian, Guochang Fang, Guilong Gui. Well-posedness problem and blow-up for an integrable shallow water equation with strong dispersive term. International Journal of Nonlinear Science 2006 (1): 3-12
[41] GM. Coclite, K.H. Karlsen, N.H. Risebro. Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, Pure Math. 2006 (6): 1-23
[42] H. Lundmark, Formation. Dynamics of shock waves in the Degasperis -Procesi equation. Journal of Nonlinear Science 2007 (17): 169-198
[43] Lixin Tian, Xiuming Li. On the well-posedness problem for the generalized Degasperis- Procesi equation. International Journal of Nonlinear Science. 2006 (2): 67-76
[44] Liqin Yu, Lixin Tian, Xuedi Wang. The bifucation and peakon for Degasperis-Procesi equation. Chaos, Soliton & Fractals 2006 (30): 956-966
[45] Lixin Tian, Chunyu Shen. Optimal control of the viscous Degasperis-Procesi equation. J. Math. Phy. 2007 (48): 13-16
[46] Lixin Tian, Xiuming Li. Well-posedness for a New Completely Integrable Shallow Water Wave Equation. International Journal of Nonlinear Science. 2007 (4): 83-91
[47] Lixin Tian, Meijie Ni. Blow-up phenomena for the periodic Degasperis -Procesi equation with strong dispersive term. International Journal of Nonlinear Science. 2006 (2): 177-182
[48] Lixin Tian, Meijie Ni. Blow-up phenomena for the Degasperis-Procesi equation with strong dispersive term. International Journal of Nonlinear Science. 2007 (3): 187-194
[49] Jiangbo Zhou, Anchan Mai. A Mixed Initial Boundary Value Problem for the Degasperis-Procesi Equation on a Bounded Interval. International Journal of Nonlinear Science. 2008 (5): 255-260
[50] Jiangbo Zhou. Global Existence of Solution to an Initial Boundary Value Problem for the Degasperis-Procesi Equation. International Journal of Nonlinear Science. 2007 (4): 141-146
[51] G. B. Whitham. Varantional methods and applications to water wave. Proc. R. Soc. Lond. A 1967 (299): 6-25
[52] B. Fornberg, G. B. Whitham. A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. R. Soc. Lond. A1978 (289): 373-404
[53] R. Ivanov. On the integrability of a class of nonlinear dispersive wave equations. J. Nonlinear Math. Phys. 2005 (1294): 462-468
[54] A. Constantin, J. Escher. Wave breaking for nonlinear nonlocal shallow water equations. Acta Mathematica 1998 (181): 229-243
[55] A. Constantin. The trajectories of particles in Stokes waves. Invent. Math. 2006 (166): 523-535
[56] J. F. Toland. Stokes waves. Topol. Methods Nonlinear Anal. 1996 (7): 1-48
[57] A. Constantin, J. Escher. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 2007 (44): 423-431
[58] Jiangbo Zhou, Lixin Tian. A type of bounded traveling wave solutions for the Fornberg -Whitham equation. J. Math. Anal. Appl. 2008 (346): 255-261
[59] Jiangbo Zhou, Lixin Tian. Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation. Nonlinear Anal: Real World Appl.doi:10. 1016/j.nonrwa.2008.11.014
[60] Babin A V, Vishik M I. Attractor of Evolution Equations. Studies in Math.and its Appl, North-Holland, Amsterdam, London, New York, Tokyo: 1992,25