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The gap lemma and geometric criteria for instability of viscous shock profiles. (English) Zbl 0933.35136

The particular interest in the paper is in viscosous shock waves, which are traveling wave solutions of systems of conservation laws \[ u_t+f(u)_x=(B(u)u_x)_x \] tending to asymptotic values \(u_{\pm}\) as \(x\to\pm\infty\). In general, traveling waves arise as stationary solutions of nonlinear systems of PDEs. If the linearized equation about such a stationary solution is \(v_t=Mv\), then the eigenvalue equation \(Mw=\lambda w\) can be recast as a nonautonomous, linear system of ODEs of the form \( W'=A(x,\lambda)W. \) The stability analysis of the traveling waves is related to the spectrum of the linearized equation, and consequently with the Evans function, defined as a Wronskian of certain solutions that decay if either \(t\to+\infty\) or \(t\to-\infty\). An obstacle in the use of the Evans function for stability analysis of traveling waves occurs when the spectrum of the linearized operator accumulates at the imaginary axis. This difficulty arises in the analysis of viscous shock profiles. In the paper a general result is proved, the “gap lemma”, concerning the analytic continuation of the Evans function. This leads to a necessary geometric condition in terms of the sign of certain integral of the associated viscous profile. Using this method the stability of certain undercompressive shock waves is analysed. Further, it is shown, that for a wide class of systems, the homoclinic (solitary) waves are linearly unstable. It is also shown that heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for \(n\times n\) conservation laws, \(n\geq 2\).

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
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