# 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 and Continuous Dynamical Systems, 2012, 32 (1) : 331-352. doi: 10.3934/dcds.2012.32.331
##### 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), 848-868. doi: 10.1007/s00033-008-7026-1. [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), 767-783. [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), 1152-1170. doi: 10.1016/j.jmaa.2005.11.002. [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), 15-35. doi: 10.1016/j.jmaa.2004.06.007. [5] P. C. Fife, Considerations regarding the mathematical basis for Prandtl's boundary layer theory, Arch. Rational Mech. Anal., 28 (1967/1968), 184-216. [6] H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids, Comm. Math. Phys., 208 (1999), 309-330. doi: 10.1007/s002200050760. [7] H. Frid and V. Shelukhin, Boundary layers in parabolic perturbations of scalar conservation laws, Z. Angew. Math. Phys., 55 (2004), 420-434. doi: 10.1007/s00033-003-1094-z. [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), 377-382. [9] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146. [10] D. Y. Hsieh, On partial differential equations related to Lorenz system, J. Math. Phys., 28 (1987), 1589-1597. doi: 10.1063/1.527465. [11] H. Y. Jian and D. G. Chen, On the Cauchy problem for certain system of semilinear parabolic equations, Acta Math. Sinica, 14 (1998), 27-34. doi: 10.1007/BF02563880. [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), 237-268. doi: 10.1137/07070005X. [13] L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation, Stud. Appl. Math., 73 (1985), 91-153. [14] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Progr. Theoret. Phys., 54 (1975), 687-699. doi: 10.1143/PTP.54.687. [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), 57-82. doi: 10.1007/s002050050047. [16] K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. angew. Math. Phys., 57 (2006), 604-614. doi: 10.1007/s00033-006-0062-9. [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), 439-458. [18] O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation," 15. Chapman & Hall/CRC, Boca Raton, FL, 1999. [19] F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3. [20] H. Schlichting and K. Gersten, "Boundary-Layer Theory," with contributions by Egon Krause and Herbert Oertel, Jr., translated from the ninth German edition by Katherine Mayes, eighth revised and enlarged edition, Springer-Verlag, Berlin, 2000. [21] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. doi: 10.1007/s002200100486. [22] S. Q. Tang and H. J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl., 233 (1999), 336-358. doi: 10.1006/jmaa.1999.6316. [23] G. Tian and Z. P. Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. [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), 1256-1298. doi: 10.1137/040614967. [25] Z.A. Wang, Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity, Z. Angew. Math. Phys., 57 (2006), 399-418. doi: 10.1007/s00033-005-0030-9. [26] Z. P. Xin, Viscous boundary layers and their stability I., J. Partial Differential Equations, 11 (1998), 97-124. [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), 479-541. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. [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), 994-1014. doi: 10.1007/s00033-004-3117-9.

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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), 848-868. doi: 10.1007/s00033-008-7026-1. [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), 767-783. [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), 1152-1170. doi: 10.1016/j.jmaa.2005.11.002. [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), 15-35. doi: 10.1016/j.jmaa.2004.06.007. [5] P. C. Fife, Considerations regarding the mathematical basis for Prandtl's boundary layer theory, Arch. Rational Mech. Anal., 28 (1967/1968), 184-216. [6] H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids, Comm. Math. Phys., 208 (1999), 309-330. doi: 10.1007/s002200050760. [7] H. Frid and V. Shelukhin, Boundary layers in parabolic perturbations of scalar conservation laws, Z. Angew. Math. Phys., 55 (2004), 420-434. doi: 10.1007/s00033-003-1094-z. [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), 377-382. [9] E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations, 143 (1998), 110-146. [10] D. Y. Hsieh, On partial differential equations related to Lorenz system, J. Math. Phys., 28 (1987), 1589-1597. doi: 10.1063/1.527465. [11] H. Y. Jian and D. G. Chen, On the Cauchy problem for certain system of semilinear parabolic equations, Acta Math. Sinica, 14 (1998), 27-34. doi: 10.1007/BF02563880. [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), 237-268. doi: 10.1137/07070005X. [13] L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation, Stud. Appl. Math., 73 (1985), 91-153. [14] Y. Kuramoto and T. Tsuzuki, On the formation of dissipative structures in reaction-diffusion systems, Progr. Theoret. Phys., 54 (1975), 687-699. doi: 10.1143/PTP.54.687. [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), 57-82. doi: 10.1007/s002050050047. [16] K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. angew. Math. Phys., 57 (2006), 604-614. doi: 10.1007/s00033-006-0062-9. [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), 439-458. [18] O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation," 15. Chapman & Hall/CRC, Boca Raton, FL, 1999. [19] F. Rousset, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems, Trans. Amer. Math. Soc., 355 (2003), 2991-3008. doi: 10.1090/S0002-9947-03-03279-3. [20] H. Schlichting and K. Gersten, "Boundary-Layer Theory," with contributions by Egon Krause and Herbert Oertel, Jr., translated from the ninth German edition by Katherine Mayes, eighth revised and enlarged edition, Springer-Verlag, Berlin, 2000. [21] D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Comm. Math. Phys., 221 (2001), 267-292. doi: 10.1007/s002200100486. [22] S. Q. Tang and H. J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl., 233 (1999), 336-358. doi: 10.1006/jmaa.1999.6316. [23] G. Tian and Z. P. Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. [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), 1256-1298. doi: 10.1137/040614967. [25] Z.A. Wang, Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity, Z. Angew. Math. Phys., 57 (2006), 399-418. doi: 10.1007/s00033-005-0030-9. [26] Z. P. Xin, Viscous boundary layers and their stability I., J. Partial Differential Equations, 11 (1998), 97-124. [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), 479-541. doi: 10.1002/(SICI)1097-0312(199904)52:4<479::AID-CPA4>3.0.CO;2-1. [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), 994-1014. doi: 10.1007/s00033-004-3117-9.
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