February  2010, 27(1): 133-169. doi: 10.3934/dcds.2010.27.133

Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks

1. 

Department of Mathematics, Penn State University, University Park, State College, PA 16802, United States, United States

2. 

Department of Mathematics, Univ. of North Carolina, Chapel Hill, NC 27599, United States

Received  May 2009 Revised  December 2009 Published  February 2010

We construct stationary solutions to the non-barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. The equation of state is assumed to satisfy standard monotonicity and convexity assumptions. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in an exterior domain. On the other hand, stationary smooth solutions in an interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres (or cylinders).
   Next we construct smooth solutions wε to the Navier-Stokes system converging to the previously constructed Euler shocks in the small viscosity limit ε  →  0. The viscous solutions are obtained by a new technique for constructing solutions to a class of two-point boundary problems with a fast transition region. The construction is explicit in the sense that it produces high order expansions in powers of ε for wε, and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. The solutions to the Euler equations described above provide the slowly varying contribution to the leading term in the expansion.
   The approach developed here is applicable to a variety of singular perturbation problems, including the construction of heteroclinic orbits with fast transitions. For example, a variant of our method is used in [W] to give a new construction of detonation profiles for the reactive Navier-Stokes equations.
Citation: Erik Endres, Helge Kristian Jenssen, Mark Williams. Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 133-169. doi: 10.3934/dcds.2010.27.133
[1]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

[2]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127

[3]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[4]

Senoussi Guesmia, Abdelmouhcene Sengouga. Some singular perturbations results for semilinear hyperbolic problems. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 567-580. doi: 10.3934/dcdss.2012.5.567

[5]

Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465

[6]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[7]

Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31

[8]

Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921

[9]

Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157

[10]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

[11]

Thomas I. Vogel. Comments on radially symmetric liquid bridges with inflected profiles. Conference Publications, 2005, 2005 (Special) : 862-867. doi: 10.3934/proc.2005.2005.862

[12]

Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 1-29. doi: 10.3934/dcdsb.2019170

[13]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

[14]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485

[15]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5125-5147. doi: 10.3934/dcds.2019208

[16]

J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905

[17]

K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546

[18]

Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263

[19]

John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273

[20]

Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

[Back to Top]