American Institute of Mathematical Sciences

Advanced
October  2004, 10(4): 837-855. doi: 10.3934/dcds.2004.10.837

The Evans function and stability criteria for degenerate viscous shock waves

Peter Howard 1, and  K. Zumbrun 2,

1. 

Department of Mathematics, Texas A&M University, College Station, TX 77845, United States
2. 

Mathematics Department, Indiana University, Bloomington, IN 47405, United States

Received  June 2002 Revised  December 2003 Published  March 2004

It is well known that the stability of certain distinguished waves arising in evolutionary PDE can be determined by the spectrum of the linear operator found by linearizing the PDE about the wave. Indeed, work over the last fifteen years has shown that spectral stability implies nonlinear stability in a broad range of cases, including asymptotically constant traveling waves in both reaction--diffusion equations and viscous conservation laws. A critical step toward analyzing the spectrum of such operators was taken in the late eighties by Alexander, Gardner, and Jones, whose Evans function (generalizing earlier work of John W. Evans) serves as a characteristic function for the above-mentioned operators. Thus far, results obtained through working with the Evans function have made critical use of the function's analyticity at the origin (or its analyticity over an appropriate Riemann surface). In the case of degenerate (or sonic) viscous shock waves, however, the Evans function is certainly not analytic in a neighborhood of the origin, and does not appear to admit analytic extension to a Riemann manifold. We surmount this obstacle by dividing the Evans function (plus related objects) into two pieces: one analytic in a neighborhood of the origin, and one sufficiently small.
Keywords: degeneracy, Conservation laws, stability, Evans function..
Mathematics Subject Classification: Primary: 35B35, 34E10, 35K12, 35P0.
Citation: Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837
[1]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[2]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[3]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[4]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[5]

Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010
[6]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[7]

Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[8]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[9]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[10]

Anupam Sen, T. Raja Sekhar. Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 931-942. doi: 10.3934/cpaa.2019045
[11]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
[12]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941
[13]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[14]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[15]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure & Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[16]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[17]

Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks & Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393
[18]

Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673
[19]

Rinaldo M. Colombo, Kenneth H. Karlsen, Frédéric Lagoutière, Andrea Marson. Special issue on contemporary topics in conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : i-ii. doi: 10.3934/nhm.2016.11.2i
[20]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences