# American Institute of Mathematical Sciences

January  2012, 32(1): 331-352. doi: 10.3934/dcds.2012.32.331

## Boundary layer for nonlinear evolution equations with damping and diffusion

 1 The Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received  July 2010 Revised  March 2011 Published  September 2011

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter $\beta$ goes to zero. We show that the boundary layer thickness is of the order $O\left(\beta^\gamma\right)$ with $0<\gamma<\frac{1}{2}$.
Citation: Lizhi Ruan, Changjiang Zhu. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
##### References:
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Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids,, Comm. Math. Phys., 208 (1999), 309. doi: 10.1007/s002200050760. Google Scholar [7] H. Frid and V. Shelukhin, Boundary layers in parabolic perturbations of scalar conservation laws,, Z. Angew. Math. Phys., 55 (2004), 420. doi: 10.1007/s00033-003-1094-z. Google Scholar [8] M. Gisclon and D. Serre, Étude des conditions aus limites pour un système strictement hyperbolique via l'approximation parabolique (French) [Study of boundary conditions for a strictly hyperbolic system via parabolic approximation],, C.R. Acad. Sci. Paris Ser. I Math., 319 (1994), 377. Google Scholar [9] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. Google Scholar [10] D. Y. Hsieh, On partial differential equations related to Lorenz system,, J. Math. Phys., 28 (1987), 1589. doi: 10.1063/1.527465. Google Scholar [11] H. Y. Jian and D. G. Chen, On the Cauchy problem for certain system of semilinear parabolic equations,, Acta Math. Sinica, 14 (1998), 27. doi: 10.1007/BF02563880. Google Scholar [12] S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry,, SIAM J. Math. Anal., 41 (2009), 237. doi: 10.1137/07070005X. Google Scholar [13] L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation,, Stud. Appl. Math., 73 (1985), 91. Google Scholar [14] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems,, Progr. Theoret. Phys., 54 (1975), 687. doi: 10.1143/PTP.54.687. Google Scholar [15] T. P. Liu and S. H. Yu, Propagation of a stationary shock layer in the presence of a boundary,, Arch. Rational Mech. Anal., 139 (1997), 57. doi: 10.1007/s002050050047. Google Scholar [16] K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity,, Z. angew. Math. Phys., 57 (2006), 604. doi: 10.1007/s00033-006-0062-9. Google Scholar [17] K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the $p$-system with linear damping,, J. Differential Equations, 156 (1999), 439. Google Scholar [18] O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation,", 15. Chapman & Hall/CRC, (1999). Google Scholar [19] F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems,, Trans. Amer. Math. Soc., 355 (2003), 2991. doi: 10.1090/S0002-9947-03-03279-3. Google Scholar [20] H. Schlichting and K. Gersten, "Boundary-Layer Theory,", with contributions by Egon Krause and Herbert Oertel, (2000). Google Scholar [21] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. doi: 10.1007/s002200100486. Google Scholar [22] S. Q. Tang and H. J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity,, J. Math. Anal. Appl., 233 (1999), 336. doi: 10.1006/jmaa.1999.6316. Google Scholar [23] G. Tian and Z. P. Xin, Gradient estimation on Navier-Stokes equations,, Comm. Anal. Geom., 7 (1999), 221. Google Scholar [24] Y. G. Wang and Z. P. Xin, Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane,, SIAM J. Math. Anal., 37 (2005), 1256. doi: 10.1137/040614967. Google Scholar [25] Z.A. Wang, Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,, Z. Angew. Math. Phys., 57 (2006), 399. doi: 10.1007/s00033-005-0030-9. Google Scholar [26] Z. P. Xin, Viscous boundary layers and their stability I.,, J. Partial Differential Equations, 11 (1998), 97. Google Scholar [27] Z. P. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane,, Comm. Pure Appl. Math., 52 (1999), 479. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. Google Scholar [28] C. J. Zhu and Z. A. Wang, Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,, Z. Angew. Math. Phys., 55 (2004), 994. doi: 10.1007/s00033-004-3117-9. Google Scholar

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##### References:
 [1] W. Allegretto, Y. P. Lin and Z. Y. Zhang, Properties of global decaying solution to the Cauchy problem of nonlinear evolution equations,, Z. Angew. Math. Phys., 59 (2008), 848. doi: 10.1007/s00033-008-7026-1. Google Scholar [2] K. M. Chen and C. J. Zhu, The zero diffusion limit for nonlinear hyperbolic system with damping and diffusion,, J. Hyperbolic Differ. Equ., 5 (2008), 767. Google Scholar [3] R. J. Duan, S. Q. Tang and C. J. Zhu, Asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant,, J. Math. Anal. Appl., 323 (2006), 1152. doi: 10.1016/j.jmaa.2005.11.002. Google Scholar [4] R. J. Duan and C. J. Zhu, Asymptotics of dissipative nonlinear evolution equations with ellipticity: Different end states,, J. Math. Anal. Appl., 303 (2005), 15. doi: 10.1016/j.jmaa.2004.06.007. Google Scholar [5] P. C. Fife, Considerations regarding the mathematical basis for Prandtl's boundary layer theory,, Arch. Rational Mech. Anal., 28 (): 184. Google Scholar [6] H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids,, Comm. Math. Phys., 208 (1999), 309. doi: 10.1007/s002200050760. Google Scholar [7] H. Frid and V. Shelukhin, Boundary layers in parabolic perturbations of scalar conservation laws,, Z. Angew. Math. Phys., 55 (2004), 420. doi: 10.1007/s00033-003-1094-z. Google Scholar [8] M. Gisclon and D. Serre, Étude des conditions aus limites pour un système strictement hyperbolique via l'approximation parabolique (French) [Study of boundary conditions for a strictly hyperbolic system via parabolic approximation],, C.R. Acad. Sci. Paris Ser. I Math., 319 (1994), 377. Google Scholar [9] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110. Google Scholar [10] D. Y. Hsieh, On partial differential equations related to Lorenz system,, J. Math. Phys., 28 (1987), 1589. doi: 10.1063/1.527465. Google Scholar [11] H. Y. Jian and D. G. Chen, On the Cauchy problem for certain system of semilinear parabolic equations,, Acta Math. Sinica, 14 (1998), 27. doi: 10.1007/BF02563880. Google Scholar [12] S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry,, SIAM J. Math. Anal., 41 (2009), 237. doi: 10.1137/07070005X. Google Scholar [13] L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation,, Stud. Appl. Math., 73 (1985), 91. Google Scholar [14] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems,, Progr. Theoret. Phys., 54 (1975), 687. doi: 10.1143/PTP.54.687. Google Scholar [15] T. P. Liu and S. H. Yu, Propagation of a stationary shock layer in the presence of a boundary,, Arch. Rational Mech. Anal., 139 (1997), 57. doi: 10.1007/s002050050047. Google Scholar [16] K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity,, Z. angew. Math. Phys., 57 (2006), 604. doi: 10.1007/s00033-006-0062-9. Google Scholar [17] K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the $p$-system with linear damping,, J. Differential Equations, 156 (1999), 439. Google Scholar [18] O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation,", 15. Chapman & Hall/CRC, (1999). Google Scholar [19] F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems,, Trans. Amer. Math. Soc., 355 (2003), 2991. doi: 10.1090/S0002-9947-03-03279-3. Google Scholar [20] H. Schlichting and K. Gersten, "Boundary-Layer Theory,", with contributions by Egon Krause and Herbert Oertel, (2000). Google Scholar [21] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. doi: 10.1007/s002200100486. Google Scholar [22] S. Q. Tang and H. J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity,, J. Math. Anal. Appl., 233 (1999), 336. doi: 10.1006/jmaa.1999.6316. Google Scholar [23] G. Tian and Z. P. Xin, Gradient estimation on Navier-Stokes equations,, Comm. Anal. Geom., 7 (1999), 221. Google Scholar [24] Y. G. Wang and Z. P. Xin, Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly oscillatory forces in the half-plane,, SIAM J. Math. Anal., 37 (2005), 1256. doi: 10.1137/040614967. Google Scholar [25] Z.A. Wang, Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,, Z. Angew. Math. Phys., 57 (2006), 399. doi: 10.1007/s00033-005-0030-9. Google Scholar [26] Z. P. Xin, Viscous boundary layers and their stability I.,, J. Partial Differential Equations, 11 (1998), 97. Google Scholar [27] Z. P. Xin and T. Yanagisawa, Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane,, Comm. Pure Appl. Math., 52 (1999), 479. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. Google Scholar [28] C. J. Zhu and Z. A. Wang, Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity,, Z. Angew. Math. Phys., 55 (2004), 994. doi: 10.1007/s00033-004-3117-9. Google Scholar
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