The Evans function and stability criteria for degenerate viscous
shock waves
The Evans function and stability criteria for degenerate viscous
shock waves
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.
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