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:
[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.

[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.

[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.

[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.

[5]

P. C. Fife, Considerations regarding the mathematical basis for Prandtl's boundary layer theory,, Arch. Rational Mech. Anal., 28 (): 184.

[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.

[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.

[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.

[9]

E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110.

[10]

D. Y. Hsieh, On partial differential equations related to Lorenz system,, J. Math. Phys., 28 (1987), 1589. 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. 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. doi: 10.1137/07070005X.

[13]

L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation,, Stud. Appl. Math., 73 (1985), 91.

[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.

[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.

[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.

[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.

[18]

O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation,", 15. Chapman & Hall/CRC, (1999).

[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.

[20]

H. Schlichting and K. Gersten, "Boundary-Layer Theory,", with contributions by Egon Krause and Herbert Oertel, (2000).

[21]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. 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. 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.

[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.

[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.

[26]

Z. P. Xin, Viscous boundary layers and their stability I.,, J. Partial Differential Equations, 11 (1998), 97.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[5]

P. C. Fife, Considerations regarding the mathematical basis for Prandtl's boundary layer theory,, Arch. Rational Mech. Anal., 28 (): 184.

[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.

[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.

[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.

[9]

E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems,, J. Differential Equations, 143 (1998), 110.

[10]

D. Y. Hsieh, On partial differential equations related to Lorenz system,, J. Math. Phys., 28 (1987), 1589. 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. 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. doi: 10.1137/07070005X.

[13]

L. R. Keefe, Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation,, Stud. Appl. Math., 73 (1985), 91.

[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.

[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.

[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.

[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.

[18]

O. A. Oleinik and V. N. Samokhin, "Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation,", 15. Chapman & Hall/CRC, (1999).

[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.

[20]

H. Schlichting and K. Gersten, "Boundary-Layer Theory,", with contributions by Egon Krause and Herbert Oertel, (2000).

[21]

D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit,, Comm. Math. Phys., 221 (2001), 267. 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. 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.

[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.

[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.

[26]

Z. P. Xin, Viscous boundary layers and their stability I.,, J. Partial Differential Equations, 11 (1998), 97.

[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.

[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.

[1]

Walter Allegretto, Yanping Lin, Zhiyong Zhang. Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations. Conference Publications, 2009, 2009 (Special) : 11-23. doi: 10.3934/proc.2009.2009.11

[2]

Mina Jiang, Changjiang Zhu. Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 887-918. doi: 10.3934/dcds.2009.23.887

[3]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257

[4]

José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027

[5]

Hongyun Peng, Lizhi Ruan, Changjiang Zhu. Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis. Kinetic & Related Models, 2012, 5 (3) : 563-581. doi: 10.3934/krm.2012.5.563

[6]

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

[7]

Matteo Bonforte, Jean Dolbeault, Matteo Muratori, Bruno Nazaret. Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods. Kinetic & Related Models, 2017, 10 (1) : 61-91. doi: 10.3934/krm.2017003

[8]

Shu Wang, Chundi Liu. Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2177-2199. doi: 10.3934/cpaa.2017108

[9]

Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

[10]

Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure & Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479

[11]

Kelei Wang. The singular limit problem in a phase separation model with different diffusion rates $^*$. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 483-512. doi: 10.3934/dcds.2015.35.483

[12]

L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799

[13]

Noriaki Yamazaki. Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems. Conference Publications, 2005, 2005 (Special) : 920-929. doi: 10.3934/proc.2005.2005.920

[14]

Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083

[15]

Jing Wang, Lining Tong. Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers. Communications on Pure & Applied Analysis, 2019, 18 (2) : 887-910. doi: 10.3934/cpaa.2019043

[16]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723

[17]

James Broda, Alexander Grigo, Nikola P. Petrov. Convergence rates for semistochastic processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 109-125. doi: 10.3934/dcdsb.2019001

[18]

Masahiro Suzuki. Asymptotic stability of a boundary layer to the Euler--Poisson equations for a multicomponent plasma. Kinetic & Related Models, 2016, 9 (3) : 587-603. doi: 10.3934/krm.2016008

[19]

Felipe Alvarez, Alexandre Cabot. Asymptotic selection of viscosity equilibria of semilinear evolution equations by the introduction of a slowly vanishing term. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 921-938. doi: 10.3934/dcds.2006.15.921

[20]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]