# American Institute of Mathematical Sciences

January  2009, 8(1): 419-433. doi: 10.3934/cpaa.2009.8.419

## On a Nested Boundary-Layer Problem

 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States 2 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  March 2008 Revised  September 2008 Published  October 2008

Nested boundary layers mean that one boundary layer lies inside another one. In this paper, we consider one such problem, namely,

$\varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0$ with $0 < x <1$, $y(0) = 1$ and $y(1) = \sqrt{e}$.

An asymptotic solution, which holds uniformly for $x\in [0,1]$, is constructed rigorously. This result also provides an explicit formula for the exponentially small leading term in the interval where the exact solution exhibits such behavior. This henomenon has never been mentioned in the existing literature.

Citation: X. Liang, Roderick S. C. Wong. On a Nested Boundary-Layer Problem. Communications on Pure & Applied Analysis, 2009, 8 (1) : 419-433. doi: 10.3934/cpaa.2009.8.419
 [1] Pau Martín, David Sauzin, Tere M. Seara. Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 301-372. doi: 10.3934/dcds.2011.31.301 [2] Jesse Goodman, Daniel Spector. Some remarks on boundary operators of Bessel extensions. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 493-509. doi: 10.3934/dcdss.2018027 [3] R. Estrada. Boundary layers and spectral content asymptotics. Conference Publications, 1998, 1998 (Special) : 242-252. doi: 10.3934/proc.1998.1998.242 [4] Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315 [5] Philippe Chartier, Ander Murua, Jesús María Sanz-Serna. A formal series approach to averaging: Exponentially small error estimates. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3009-3027. doi: 10.3934/dcds.2012.32.3009 [6] Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213 [7] Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785 [8] Karsten Matthies. Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 585-602. doi: 10.3934/dcds.2003.9.585 [9] Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757 [10] Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez. Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies. Electronic Research Announcements, 2014, 21: 41-61. doi: 10.3934/era.2014.21.41 [11] C. E. Kenig, S. N. Ziesler. Maximal function estimates with applications to a modified Kadomstev-Petviashvili equation. Communications on Pure & Applied Analysis, 2005, 4 (1) : 45-91. doi: 10.3934/cpaa.2005.4.45 [12] Maolin Cheng, Mingyin Xiang. Application of a modified CES production function model based on improved firefly algorithm. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019018 [13] Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042 [14] Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271 [15] Makram Hamouda, Chang-Yeol Jung, Roger Temam. Boundary layers for the 2D linearized primitive equations. Communications on Pure & Applied Analysis, 2009, 8 (1) : 335-359. doi: 10.3934/cpaa.2009.8.335 [16] Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595 [17] Niclas Bernhoff. Boundary layers and shock profiles for the discrete Boltzmann equation for mixtures. Kinetic & Related Models, 2012, 5 (1) : 1-19. doi: 10.3934/krm.2012.5.1 [18] Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885 [19] Giuseppina Autuori, Patrizia Pucci. Kirchhoff systems with nonlinear source and boundary damping terms. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1161-1188. doi: 10.3934/cpaa.2010.9.1161 [20] Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069

2018 Impact Factor: 0.925