Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

The Evans function and stability criteria for degenerate viscous shock waves

Pages: 837 - 855, Volume 10, Issue 4, June 2004      doi:10.3934/dcds.2004.10.837

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Peter Howard - Department of Mathematics, Texas A&M University, College Station, TX 77845, United States (email)
K. Zumbrun - Mathematics Department, Indiana University, Bloomington, IN 47405, United States (email)

Abstract: 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:  Conservation laws, stability, degeneracy, Evans function.
Mathematics Subject Classification:  Primary: 35B35, 34E10, 35K12, 35P05.

Received: June 2002;      Revised: December 2003;      Available Online: March 2004.