Pointwise estimates and stability for dispersive-diffusive shock
waves
Pointwise estimates and stability for dispersive-diffusive shock
waves
We study the stability and pointwise behavior of perturbed
viscous shock waves for a general scalar conservation law with
constant diffusion and dispersion. Along with the usual Lax
shocks, such equations are known to admit undercompressive
shocks. We unify the treatment of these two cases by
introducing a new wave-tracking method based on
``instantaneous projection," giving improved estimates
even in the Lax case. Another important feature connected
with the introduction of dispersion is the treatment of
a non-sectorial operator. An immediate consequence of
our pointwise estimates is a simple spectral criterion for
stability in all L^p norms, p >= 1 for the Lax case and
p > 1 for the undercompressive case.
Our approach extends immediately to the case of certain
scalar equations of higher order, and would also appear
suitable for systems.
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