Differential and Integral Equations

Nonlinear stability of degenerate shock profiles

Peter Howard

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We consider degenerate viscous shock profiles arising in systems of two regularized conservation laws, where degeneracy here describes viscous shock profiles for which the asymptotic endstates are sonic to the associated hyperbolic system (the shock speed is equal to one of the characteristic speeds). Proceeding with pointwise estimates on the Green's function for the linear system of equations that arises upon linearization of the conservation law about a degenerate viscous shock profile, we establish that spectral stability, defined in terms of an Evans function, implies nonlinear stability. The asymptotic rate of decay for the perturbation is found both pointwise and in all $L^p$ norms, $p \ge 1$.

Article information

Differential Integral Equations Volume 20, Number 5 (2007), 515-560.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L65: Conservation laws
Secondary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations 35L67: Shocks and singularities [See also 58Kxx, 76L05]


Howard, Peter. Nonlinear stability of degenerate shock profiles. Differential Integral Equations 20 (2007), no. 5, 515--560. https://projecteuclid.org/euclid.die/1356039442.

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