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A BKM's criterion of smooth solution to the incompressible viscoelastic flow
Convergence rate to strong boundary layer solutions for generalized BBMBurgers equations with nonconvex 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 
References:
[1] 
H. P. Cui, The asymptotic behavior of solutions of an initial boundary value problem for the generalized BenjaminBonaMahony equation, Indian J. Pure Appl. Math., 43 (2012), 323342. doi: 10.1007/s1322601200205. 
[2] 
E. Grenier and F. Rousset, Stability of onedimensional boundary layers by using Green's functions, Communications on Pure and Applied Mathematics, 54 (2001), 13431385. doi: DOI: 10.1002/cpa.10006. 
[3] 
Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible NavierStokes equation on the half space, Commun. Math. Phys., 266 (2006), 401430. doi: 10.1007/s0022000600171. 
[4] 
S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for onedimensional gas motion, Commun. Math. Phys., 101 (1985), 97127. doi: 10.1007/BF01212358. 
[5] 
S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with nonconvex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 15471569. doi: 10.1002/cpa.3160471202. 
[6] 
S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for twodimensional viscous conservation laws in half plane, Discrete and Continuous Dynamical Systems, Supplement (2003), 469476. 
[7] 
S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multidimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581603. 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), 296320. doi: 10.1006/jdeq.1996.3217. 
[9] 
A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with nonconvex nonlinearity, Commun. Math. Phys., 165 (1994), 8396. doi: 10.1007/BF02099739. 
[10] 
T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible NavierStokes equation in a half line, J. Differential Equations, 241 (2007), 94111. 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), 107132. 
[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), 329343. 
[13] 
H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized BenjaminBonaMahonyBurgers equations in the halfspace, Kinetic and Related Models, 2 (2009), 31443216. 
[14] 
H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized BenjaminBonaMahonyBurgers equations in the halfspace, J. Differential Equations, 245 (2008), 31443216. 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), 273306. doi: 10.1006/jdeq.2001.4063. 
[16] 
P. C. Zhu, Nonlinear Waves for the Compressible NavierStokes 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 BenjaminBonaMahony equation, Indian J. Pure Appl. Math., 43 (2012), 323342. doi: 10.1007/s1322601200205. 
[2] 
E. Grenier and F. Rousset, Stability of onedimensional boundary layers by using Green's functions, Communications on Pure and Applied Mathematics, 54 (2001), 13431385. doi: DOI: 10.1002/cpa.10006. 
[3] 
Y. Kagei and S. Kawashima, Stability of planar stationary solutions to the compressible NavierStokes equation on the half space, Commun. Math. Phys., 266 (2006), 401430. doi: 10.1007/s0022000600171. 
[4] 
S. Kawashima and A. Matsumura, Asymptotic stability of travelling wave solutions of systems for onedimensional gas motion, Commun. Math. Phys., 101 (1985), 97127. doi: 10.1007/BF01212358. 
[5] 
S. Kawashima and A. Matsumura, Stability of shock profiles in viscoelasticity with nonconvex constitutive relations, Commun. Pure Appl. Math., 47 (1994), 15471569. doi: 10.1002/cpa.3160471202. 
[6] 
S. Kawashima, S. Nishibata and M. Nishikawa, Asymptotic stability of stationary waves for twodimensional viscous conservation laws in half plane, Discrete and Continuous Dynamical Systems, Supplement (2003), 469476. 
[7] 
S. Kawashima, S. Nishibata and M. Nishikawa, $L^p$ energy method for multidimensional viscous coonservation laws and applications to the stability of planar waves, J. Hyperbolic Differential Equations, 1 (2004), 581603. 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), 296320. doi: 10.1006/jdeq.1996.3217. 
[9] 
A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with nonconvex nonlinearity, Commun. Math. Phys., 165 (1994), 8396. doi: 10.1007/BF02099739. 
[10] 
T. Nakamura, S. Nishibata and, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible NavierStokes equation in a half line, J. Differential Equations, 241 (2007), 94111. 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), 107132. 
[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), 329343. 
[13] 
H. Yin and H. J. Zhao, Nonlinear stability of boundary layer solutions for generalized BenjaminBonaMahonyBurgers equations in the halfspace, Kinetic and Related Models, 2 (2009), 31443216. 
[14] 
H. Yin, H. J. Zhao and J. S. Kim, Convergence rates of solutions toward boundary layer solutions for generalized BenjaminBonaMahonyBurgers equations in the halfspace, J. Differential Equations, 245 (2008), 31443216. 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), 273306. doi: 10.1006/jdeq.2001.4063. 
[16] 
P. C. Zhu, Nonlinear Waves for the Compressible NavierStokes Equations in the Half Space, the report for JSPS postdoctoral research at Kyushu University, Fukuoka, Japan, August 2001. 
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