Nonlinear stability of degenerate shock profiles
Nonlinear stability of degenerate shock profiles
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, $p \ge 1$.
Return to Publications page