# American Institute of Mathematical Sciences

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 $u_b Citation: Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835 ##### References:  [1] H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized Benjamin-Bona-Mahony equation,, Indian J. Pure Appl. Math., 43 (2012), 323. doi: 10.1007/s13226-012-0020-5. Google Scholar [2] E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions,, Communications on Pure and Applied Mathematics, 54 (2001), 1343. doi: DOI: 10.1002/cpa.10006. Google Scholar [3] Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space,, Commun. Math. Phys., 266 (2006), 401. doi: 10.1007/s00220-006-0017-1. Google Scholar [4] S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion,, Commun. Math. Phys., 101 (1985), 97. doi: 10.1007/BF01212358. Google Scholar [5] S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations,, Commun. Pure Appl. Math., 47 (1994), 1547. doi: 10.1002/cpa.3160471202. Google Scholar [6] S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane,, Discrete and Continuous Dynamical Systems, Supplement (2003), 469. Google Scholar [7] S. Kawashima, S. Nishibata and M. Nishikawa,$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] Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 [2] Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 [3] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [4] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [5] Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020210 [6] Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329 [7] Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027 [8] Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405 [9] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [10] Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 [11] Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 [12] Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with$\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 [13] Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 [14] Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 [15] Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 [16] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of$R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 [17] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for$S^1\$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [18] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [19] Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995 [20] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

2019 Impact Factor: 1.105