
Previous Article
On the existence of solutions for the NavierStokes system in a sum of weak$L^{p}$ spaces
 DCDS Home
 This Issue

Next Article
Ground states of the SchrödingerMaxwell system with dirac mass: Existence and asymptotics
Singularly perturbed ODEs and profiles for stationary symmetric Euler and NavierStokes 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 
Next we construct smooth solutions w^{ε} to the NavierStokes 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 twopoint 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 w^{ε}, 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 NavierStokes equations.
[1] 
K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624633. doi: 10.3934/proc.2007.2007.624 
[2] 
Feliz Minhós, A. I. Santos. Higher order twopoint boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems  S, 2008, 1 (1) : 127137. 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) : 179193. 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) : 567580. doi: 10.3934/dcdss.2012.5.567 
[5] 
Jerry L. Bona, Hongqiu Chen, ShuMing Sun, BingYu Zhang. Comparison of quarterplane and twopoint boundary value problems: The KdVequation. Discrete & Continuous Dynamical Systems  B, 2007, 7 (3) : 465495. 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) : 143164. doi: 10.3934/jcd.2015001 
[7] 
XiaoYu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear twopoint boundary value problems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 3143. doi: 10.3934/naco.2012.2.31 
[8] 
Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.Y. Zhang. Comparison of quarterplane and twopoint boundary value problems: the BBMequation. Discrete & Continuous Dynamical Systems  A, 2005, 13 (4) : 921940. 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) : 11571178. 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) : 125. doi: 10.3934/eect.2020014 
[11] 
Thomas I. Vogel. Comments on radially symmetric liquid bridges with inflected profiles. Conference Publications, 2005, 2005 (Special) : 862867. doi: 10.3934/proc.2005.2005.862 
[12] 
Christian Lax, Sebastian Walcher. Singular perturbations and scaling. Discrete & Continuous Dynamical Systems  B, 2020, 25 (1) : 129. doi: 10.3934/dcdsb.2019170 
[13] 
ChanGyun Kim, YongHoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834843. doi: 10.3934/proc.2011.2011.834 
[14] 
Wenming Zou. Multiple solutions results for twopoint boundary value problem with resonance. Discrete & Continuous Dynamical Systems  A, 1998, 4 (3) : 485496. doi: 10.3934/dcds.1998.4.485 
[15] 
Ángela JiménezCasas, Aníbal RodríguezBernal. Boundary feedback as a singular limit of damped hyperbolic problems with terms concentrating at the boundary. Discrete & Continuous Dynamical Systems  A, 2019, 39 (9) : 51255147. 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) : 905915. 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) : 546555. doi: 10.3934/proc.2005.2005.546 
[18] 
Wenying Feng. Solutions and positive solutions for some threepoint boundary value problems. Conference Publications, 2003, 2003 (Special) : 263272. doi: 10.3934/proc.2003.2003.263 
[19] 
John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multipoint boundary value problems. Conference Publications, 2013, 2013 (special) : 273281. doi: 10.3934/proc.2013.2013.273 
[20] 
Lingju Kong, Qingkai Kong. Existence of nodal solutions of multipoint boundary value problems. Conference Publications, 2009, 2009 (Special) : 457465. doi: 10.3934/proc.2009.2009.457 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]