河床流体模型方程扭状孤波解的渐近稳定性
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摘要
河床流体模型方程是出现在两相流体动力学中的重要模型,本文研究了该模型单调递减扭状孤波解的渐近稳定性.文中我们首先推导了关于该扭状孤波解的一阶、二阶导数估计,然后再运用恰当的能量估计技巧和Young不等式,克服了该模型复杂耗散项引起的困难,得到了其扭状孤波解关于扰动的一致能量估计,从而证明了该模型单调递减扭状孤波解的渐近稳定性.
The ?uidized-bed modeling equation is an important model in the dynamics of twophase ?ow. In this paper, we consider the asymptotic stability of the monotone decreasing kink pro?le solitary-wave solution of the equation. At ?rst, we obtain the ?rst and second derivative estimates of the kink pro?le solitary-wave solution. Then according to the technical energy estimation and Young inequalities, we overcome the di?culty caused by the complex dissipative term, and establish the uniformly energy estimate for the perturbation of the traveling wave solution. Finally, we prove that the monotone decreasing kink pro?le solitary-wave solution is asymptotically stable.
The ?uidized-bed modeling equation is an important model in the dynamics of twophase ?ow. In this paper, we consider the asymptotic stability of the monotone decreasing kink pro?le solitary-wave solution of the equation. At ?rst, we obtain the ?rst and second derivative estimates of the kink pro?le solitary-wave solution. Then according to the technical energy estimation and Young inequalities, we overcome the di?culty caused by the complex dissipative term, and establish the uniformly energy estimate for the perturbation of the traveling wave solution. Finally, we prove that the monotone decreasing kink pro?le solitary-wave solution is asymptotically stable.
引文
[1] Ganser G H, Drew D A. Nonlinear periodic waves in a two-phase?ow model[J]. SIAM Journal on Applied Mathematics, 1987, 47(4):726-736
[2] Ganser G H, Drew D A. Nonlinear stability analysis of a uniformly?uidized bed[J]. International Journal of Multiphase Flow, 1990, 16(3):447-460
[3] Abia L, Christie I, Sanz-Serna J M. Stability of schemes for the numerical treatment of an equation modelling?uidized beds[J]. Mathematical Modelling and Numerical Analysis, 1989, 23(2):191-204
[4] Lopez-Marcos J C, Sanz-Serna J M. A de?nition of stability for nonlinear equations in numerical treat of di?erential equation[J]. Proceeding of the Forth Seminar Numerical Di?erential, 1988, 104:216-226
[5]冯民富,明平兵.河床流体模型方程的稳定化有限差分法及非线性稳定性分析[J].高等学校计算数学学报,1997,19(4):298-311Feng M F, Ming P B. A stabilized?nite di?erence method for the equation of modelling?uidized bed and it’s analysis of nonlinear stability[J]. Numerical Mathematics:A Journal of Chinese Universities, 1997,19(4):298-311
[6] Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J]. Philosophical Magazine, 1895, 39(240):422-443
[7] Bona J L, Schonbek M E. Travelling-wave solutions to the Korteweg-de Vries-Burgers equation[J]. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 1985, 101(3-4):207-226
[8] Canosa J, Gazdag J. The Korteweg-de Vries-Burgers equation[J]. Journal of Computational Physics, 1977,23(4):393-403
[9] Demiray H. A note on the exact travelling wave solution to the KdV-Burgers equation[J]. Wave Motion,2003, 38(4):367-369
[10] Feng Z S. The?rst-integral method to study the Burgers-Korteweg-de Vries equation[J]. Journal of Physics A:Mathematical and General, 2002, 35(2):343-349
[11] Guan K Y, Gao G. Qualitative research of traveling wave solution of mixed Burgers-KdV equation[J]. Acta Scientia Sinica, 1987, 17(1):64-73
[12] Xiong S L. An analytic solution of Burgers-KdV equation[J]. Chinese Science Bulletin, 1989, 34(14):1158-1162
[13] Zhang W G, Bian L, Zhao Y. Qualitative analysis and solutions of bounded travelling waves for the?uidizedbed modelling equation[J]. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 2010,140(2):241-257
[14] Goodman J. Nonlinear asymptotic stability of viscous shock pro?les for conservation laws[J]. Archive for Rational Mechanics and Analysis, 1986, 95(4):325-344
[15] Pego R L. Remarks on the stability of shock pro?les for conservation laws with dissipation[J]. Transactions of the American Mathematical Society, 1985, 291(1):353-361
[16] Matsumura A, Nishihara K. Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity[J]. Communications in Mathematical Physics, 1994, 165(1):83-96
[17] Li T, Wang Z A. Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis[J]. Journal of Di?erential Equations, 2011, 250(3):1310-1333
[18] Jin H Y, Li J, Wang Z A. Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity[J]. Journal of Di?erential Equations, 2013, 255(2):193-219
[19] Zhang W, Li X, Yong Y. Asymptotic stability of monotone increasing traveling wave solutions for viscous compressible?uid equations with capillarity term[J]. Journal of Mathematical Analysis and Applications,2016, 434(1):401-412
[20] Evans L C. Partial Di?erential Equations[M]. Providence:American Mathematical Society, 2010
[21]张愿章. Young不等式的证明及应用[J].河南科学,2004, 22(1):23-29Zhang Y Z. The proof and expansion of Young inequality[J]. Henan Science, 2004, 22(1):23-29
[2] Ganser G H, Drew D A. Nonlinear stability analysis of a uniformly?uidized bed[J]. International Journal of Multiphase Flow, 1990, 16(3):447-460
[3] Abia L, Christie I, Sanz-Serna J M. Stability of schemes for the numerical treatment of an equation modelling?uidized beds[J]. Mathematical Modelling and Numerical Analysis, 1989, 23(2):191-204
[4] Lopez-Marcos J C, Sanz-Serna J M. A de?nition of stability for nonlinear equations in numerical treat of di?erential equation[J]. Proceeding of the Forth Seminar Numerical Di?erential, 1988, 104:216-226
[5]冯民富,明平兵.河床流体模型方程的稳定化有限差分法及非线性稳定性分析[J].高等学校计算数学学报,1997,19(4):298-311Feng M F, Ming P B. A stabilized?nite di?erence method for the equation of modelling?uidized bed and it’s analysis of nonlinear stability[J]. Numerical Mathematics:A Journal of Chinese Universities, 1997,19(4):298-311
[6] Korteweg D J, de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[J]. Philosophical Magazine, 1895, 39(240):422-443
[7] Bona J L, Schonbek M E. Travelling-wave solutions to the Korteweg-de Vries-Burgers equation[J]. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 1985, 101(3-4):207-226
[8] Canosa J, Gazdag J. The Korteweg-de Vries-Burgers equation[J]. Journal of Computational Physics, 1977,23(4):393-403
[9] Demiray H. A note on the exact travelling wave solution to the KdV-Burgers equation[J]. Wave Motion,2003, 38(4):367-369
[10] Feng Z S. The?rst-integral method to study the Burgers-Korteweg-de Vries equation[J]. Journal of Physics A:Mathematical and General, 2002, 35(2):343-349
[11] Guan K Y, Gao G. Qualitative research of traveling wave solution of mixed Burgers-KdV equation[J]. Acta Scientia Sinica, 1987, 17(1):64-73
[12] Xiong S L. An analytic solution of Burgers-KdV equation[J]. Chinese Science Bulletin, 1989, 34(14):1158-1162
[13] Zhang W G, Bian L, Zhao Y. Qualitative analysis and solutions of bounded travelling waves for the?uidizedbed modelling equation[J]. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 2010,140(2):241-257
[14] Goodman J. Nonlinear asymptotic stability of viscous shock pro?les for conservation laws[J]. Archive for Rational Mechanics and Analysis, 1986, 95(4):325-344
[15] Pego R L. Remarks on the stability of shock pro?les for conservation laws with dissipation[J]. Transactions of the American Mathematical Society, 1985, 291(1):353-361
[16] Matsumura A, Nishihara K. Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity[J]. Communications in Mathematical Physics, 1994, 165(1):83-96
[17] Li T, Wang Z A. Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis[J]. Journal of Di?erential Equations, 2011, 250(3):1310-1333
[18] Jin H Y, Li J, Wang Z A. Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity[J]. Journal of Di?erential Equations, 2013, 255(2):193-219
[19] Zhang W, Li X, Yong Y. Asymptotic stability of monotone increasing traveling wave solutions for viscous compressible?uid equations with capillarity term[J]. Journal of Mathematical Analysis and Applications,2016, 434(1):401-412
[20] Evans L C. Partial Di?erential Equations[M]. Providence:American Mathematical Society, 2010
[21]张愿章. Young不等式的证明及应用[J].河南科学,2004, 22(1):23-29Zhang Y Z. The proof and expansion of Young inequality[J]. Henan Science, 2004, 22(1):23-29