American Institute of Mathematical Sciences

Advanced
March  2014, 13(2): 835-858. doi: 10.3934/cpaa.2014.13.835

Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux

Tong Li 1, and  Hui Yin 2,

1. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received  June 2013 Revised  August 2013 Published  October 2013

This paper is concerned with the initial-boundary value problem for the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $R_+$ \begin{eqnarray} u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\ u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\ u(t,0)=u_b. \end{eqnarray} Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+\not=u_b$ are two given constant states and the nonlinear function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b
Keywords: convergence rate., global stability, Generalized Benjamin-Bona-Mahony-Burgers equation, non-convex flux, boundary layer solution.
Mathematics Subject Classification: Primary: 35Q53, 35B35; Secondary: 76B1.
Citation: Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835
References:
[1]

H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation,, Indian J. Pure Appl. Math., 43 (2012), 323.  doi: 10.1007/s13226-012-0020-5.  Google Scholar

[2]

E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions,, Communications on Pure and Applied Mathematics, 54 (2001), 1343.  doi: DOI: 10.1002/cpa.10006.  Google Scholar

[3]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space,, Commun. Math. Phys., 266 (2006), 401.  doi: 10.1007/s00220-006-0017-1.  Google Scholar

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion,, Commun. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar

[5]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations,, Commun. Pure Appl. Math., 47 (1994), 1547.  doi: 10.1002/cpa.3160471202.  Google Scholar

[6]

S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane,, Discrete and Continuous Dynamical Systems, Supplement (2003), 469.   Google Scholar

[7]

S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves,, J. Hyperbolic Differential Equations, 1 (2004), 581.  doi: 10.1142/S0219891604000196.  Google Scholar

[8]

T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect,, J. Differential Equations, 133 (1997), 296.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[9]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity,, Commun. Math. Phys., 165 (1994), 83.  doi: 10.1007/BF02099739.  Google Scholar

[10]

T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differential Equations, 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[11]

M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws,, Funk. Ekvac., 41 (1998), 107.   Google Scholar

[12]

Y. Ueda, Asymptotic convergence toward stationary waves to the wave equation with a convection term in the half space,, Advances in Mathematical Sciences and Applications, 18 (2008), 329.   Google Scholar

[13]

H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space,, Kinetic and Related Models, 2 (2009), 3144.   Google Scholar

[14]

H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space,, J. Differential Equations, 245 (2008), 3144.  doi: 10.1016/j.jde.2007.12.012.  Google Scholar

[15]

C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation,, J. Differential Equations, 180 (2002), 273.  doi: 10.1006/jdeq.2001.4063.  Google Scholar

[16]

P. C. Zhu, Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space,, the report for JSPS postdoctoral research at Kyushu University, (2001).   Google Scholar

show all references

References:
[1]

H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation,, Indian J. Pure Appl. Math., 43 (2012), 323.  doi: 10.1007/s13226-012-0020-5.  Google Scholar

[2]

E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions,, Communications on Pure and Applied Mathematics, 54 (2001), 1343.  doi: DOI: 10.1002/cpa.10006.  Google Scholar

[3]

Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space,, Commun. Math. Phys., 266 (2006), 401.  doi: 10.1007/s00220-006-0017-1.  Google Scholar

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion,, Commun. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar

[5]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations,, Commun. Pure Appl. Math., 47 (1994), 1547.  doi: 10.1002/cpa.3160471202.  Google Scholar

[6]

S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane,, Discrete and Continuous Dynamical Systems, Supplement (2003), 469.   Google Scholar

[7]

S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multi-dimensional viscous coonservation laws and applications to the stability of planar waves,, J. Hyperbolic Differential Equations, 1 (2004), 581.  doi: 10.1142/S0219891604000196.  Google Scholar

[8]

T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect,, J. Differential Equations, 133 (1997), 296.  doi: 10.1006/jdeq.1996.3217.  Google Scholar

[9]

A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity,, Commun. Math. Phys., 165 (1994), 83.  doi: 10.1007/BF02099739.  Google Scholar

[10]

T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line,, J. Differential Equations, 241 (2007), 94.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[11]

M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws,, Funk. Ekvac., 41 (1998), 107.   Google Scholar

[12]

Y. Ueda, Asymptotic convergence toward stationary waves to the wave equation with a convection term in the half space,, Advances in Mathematical Sciences and Applications, 18 (2008), 329.   Google Scholar

[13]

H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space,, Kinetic and Related Models, 2 (2009), 3144.   Google Scholar

[14]

H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equations in the half-space,, J. Differential Equations, 245 (2008), 3144.  doi: 10.1016/j.jde.2007.12.012.  Google Scholar

[15]

C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation,, J. Differential Equations, 180 (2002), 273.  doi: 10.1006/jdeq.2001.4063.  Google Scholar

[16]

P. C. Zhu, Nonlinear Waves for the Compressible Navier-Stokes Equations in the Half Space,, the report for JSPS postdoctoral research at Kyushu University, (2001).   Google Scholar

[1]

Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic & Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521
[2]

Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
[3]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[4]

Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042
[5]

C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763
[6]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171
[7]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[8]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905
[9]

Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092
[10]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[11]

Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047
[12]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[13]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[14]

. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks & Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127
[15]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134
[16]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[17]

Zigen Ouyang, Chunhua Ou. Global stability and convergence rate of traveling waves for a nonlocal model in periodic media. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 993-1007. doi: 10.3934/dcdsb.2012.17.993
[18]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004
[19]

Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032
[20]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks & Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences