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

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
Citation: Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and 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), 323-342. doi: 10.1007/s13226-012-0020-5.

[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), 1343-1385. doi: DOI: 10.1002/cpa.10006.

[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), 401-430. doi: 10.1007/s00220-006-0017-1.

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[5]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202.

[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), 469-476.

[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), 581-603. doi: 10.1142/S0219891604000196.

[8]

T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320. doi: 10.1006/jdeq.1996.3217.

[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), 83-96. doi: 10.1007/BF02099739.

[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), 94-111. doi: 10.1016/j.jde.2007.06.016.

[11]

M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws, Funk. Ekvac., 41 (1998), 107-132.

[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), 329-343.

[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), 3144-3216.

[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), 3144-3216. doi: 10.1016/j.jde.2007.12.012.

[15]

C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063.

[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, Fukuoka, Japan, August 2001.

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), 323-342. doi: 10.1007/s13226-012-0020-5.

[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), 1343-1385. doi: DOI: 10.1002/cpa.10006.

[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), 401-430. doi: 10.1007/s00220-006-0017-1.

[4]

S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion, Commun. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[5]

S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 1547-1569. doi: 10.1002/cpa.3160471202.

[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), 469-476.

[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), 581-603. doi: 10.1142/S0219891604000196.

[8]

T. P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous conservation laws with boundary effect, J. Differential Equations, 133 (1997), 296-320. doi: 10.1006/jdeq.1996.3217.

[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), 83-96. doi: 10.1007/BF02099739.

[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), 94-111. doi: 10.1016/j.jde.2007.06.016.

[11]

M. Nishikawa, Convergence rates to the travelling wave for viscous conservation laws, Funk. Ekvac., 41 (1998), 107-132.

[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), 329-343.

[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), 3144-3216.

[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), 3144-3216. doi: 10.1016/j.jde.2007.12.012.

[15]

C. J. Zhu, Asymptotic behavior of solutions for $p$-system with relaxation, J. Differential Equations, 180 (2002), 273-306. doi: 10.1006/jdeq.2001.4063.

[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, Fukuoka, Japan, August 2001.

[1]

Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092

[10]

Qiangheng Zhang. Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021293

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (91)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]