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

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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

Abstract
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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 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.

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