BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

doi:10.1016/S0252-9602(14)60084-9

摘要/Abstract

摘要：

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter α goes to zero.

关键词: nonlinear evolution equations, vanishing diffusion limit, convergence rates, boundary layer, BL-thickness

Abstract:

Key words: nonlinear evolution equations, vanishing diffusion limit, convergence rates, boundary layer, BL-thickness

中图分类号:

35K45

彭艳. BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(4): 1271-1286.

PENG Yan. BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS[J]. Acta mathematica scientia,Series B, 2014, 34(4): 1271-1286.

参考文献

[1] Allegretto W, Lin Y P, Zhang Z Y. Asymptotic behavior of solution to nonlinear evolution equations with damping. J Math Anal Appl, 2008, 347(1): 344–353

[2] Chen K M, Zhu C J. The zero diffusion limit for nonlinear hyperbolic system with damping and diffusion. J Hyperbolic Differ Equ, 2008, 5(4): 767–783

[3] Duan R J, Tang S Q, Zhu C J. Asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant. J Math Anal Appl, 2006, 323(2): 1152–1170

[4] Duan R J, Zhu C J. Asymptotics of dissipative nonlinear evolution equations with ellipticity: Different end states. J Math Anal Appl, 2005, 303(1): 15–35

[5] Duan Z W, Han S X, Zhou L. Boundary layer asmptotic behavior of incompressible Navier-Stokes equation in a cylinder with small viscosity. Acta Math Sci, 2008, 28B(3): 449–468

[6] File P C. Considerations regarding the mathematical basis for Prandtl’s boundary layer theory. Arch Rational Mech Anal, 1968, 28(3): 184–216

[7] Frid H, Shelukhin V. Boundary layers for the Navier-Stokes equations of compressible fluids. Comm Math Phys, 1999, 208(2): 309–330

[8] Frid H, Shelukhin V. Boundary layers in parabolic perturbations of scalar conservation laws. Z Angew Math Phys, 2004, 55(3): 420–434

[9] Gisclon M, Serre D. Etude des conditions aus limites pour un systeme strictement hyperbolique via l’approximation parabolique. C R Acad Sci Paris Ser I Math, 1994, 319(4): 377–382

[10] Grenier E, Gues O. Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J Differential Equations, 1998, 143(1): 110–146

[11] Hsieh D Y. On partial differential equations related to Lorenz system. J Math Phys, 1987, 28(7): 1589–1597

[12] Jian H Y, Chen D G. On the Cauchy problem for certain system of semilinear parabolic equations. Acta Math Sinica, 1998, 14(1): 27–34

[13] Jiang S, Zhang J W. Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry. SIAM J Math Anal, 2009, 41(1): 237–268

[14] Keefe L R. Dynamics of perturbed wavetrain solutions to the Ginzberg-Landau equation. Stud Appl Math, 1985, 73(2): 91–152

[15] Kuramoto Y, Tsuzuki T. On the formation of dissipative structures in reaction-diffusion systems. Progr Theoret Phys, 1975, 54(3): 687–699

[16] Liu T P, Yu S H. Propagation of a stationary shock layer in the presence of a boundary. Arch Rational Mech Anal, 1997, 139(1): 57–82

[17] Nishihara K. Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity. Z Angew Math Phys, 2006, 57(4): 604–614

[18] Nishihara K, Yang T. Boundary effect on asymptotic behavior of solutions to the p-system with linear damping. J Differential Equations, 1999, 156(2): 439–458

[19] Oleinik O A, Samokhin V N. Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation. Chapman & Hall/CRC, 1999

[20] Qin X H. Boundary layer solution for p-system with artificial viscosity. Acta Math Sci, 2009, 29B(5): 1233–1240

[21] Rousset F, Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems. Trans Amer Math Soc, 2003, 355(7): 2991–3008

[22] Ruan L Z, Zhu C J. Boundary layer for nonlinear evolution equations with damping and diffusion. Discrete Contin Dyn Syst Ser A, 2012, 32(1): 331–352

[23] Schlichting H. Boundary Layer Theory. 7th ed. London, New York: McGraw-Hill Company, 1987

[24] Serre D, Zumbrun K. Boundary layer stability in real vanishing viscosity limit. Comm Math Phys, 2001, 221(2): 267–292

[25] Tang S Q, Zhao H J. Nonlinear stability for dissipative nonlinear evolution with ellipticity. J Math Anal App, 1999, 233(1): 336–358

[26] Tian G, Xin Z P. Gradient estimation on Navier-Stokes equations. Comm Anal Geom, 1999, 7(2): 221–257

[27] Wang Y G, Xin Z P. Zero-viscosity limit of the linearized compressible Navier-Stokes equations with highly
oscillatory forces in the half-plane. SIAM J Math Anal, 2005, 37(4): 1256–1298

[28] Wang Z A. Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity. Z Angew Math Phys, 2006, 57(3): 399–418

[29] Xin Z P. Viscous boundary layers and their stability I. J Partial Differential Equations, 1998, 11(2): 97–124

[30] Xin Z P, Yanagisawa T. Zero-viscosity limit of the linearized Navier-Stokes equations for a compressible viscous fluid in the half-plane. Comm Pure Appl Math, 1999, 52(4): 479–541

[31] Zhu C J, Wang Z A. Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity. Z Angew Math Phys, 2004, 55(6): 994–1014