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DCDS

This is an expository paper whose goal is to provide a detailed survey without the full technicalities of the methods used recently in [GMWZ1, GMWZ2] to prove the existence of curved multi-D viscous shocks, to rigorously justify the small viscosity limit, and to prove long time stability of multidimensional planar viscous shocks.

KRM

Using a weighted $H^s$-contraction mapping argument based on the
macro-micro decomposition of Liu and Yu, we give an elementary proof
of existence, with sharp rates of decay and distance from the
Chapman-Enskog approximation, of small-amplitude shock profiles of
the Boltzmann equation with hard-sphere potential, recovering and
slightly sharpening results obtained by Caflisch and Nicolaenko
using different techniques. A key technical point in both analyses
is that the linearized collision operator $L$ is negative definite
on its range, not only in the standard square-root Maxwellian
weighted norm for which it is self-adjoint, but also in norms with
nearby weights. Exploring this issue further, we show that $L$ is
negative definite on its range in a much wider class of norms
including norms with weights asymptotic nearly to a full Maxwellian
rather than its square root. This yields sharp localization in
velocity at near-Maxwellian rate, rather than the square-root rate
obtained in previous analyses.

DCDS

In this paper we prove the continuity of stable subspaces associated to
parabolic-hyperbolic boundary value problems, for limiting values of
parameters. The analysis is based on
the construction performed in [MZ] of
Kreiss' type symmetrizers.

DCDS

In recent work, the second author and various collaborators have shown
using Evans function/refined semigroup techniques that, under very
general circumstances, the problems of determining one- or
multi-dimensional nonlinear stability of a smooth shock profile may be
reduced to that of determining spectral stability of the corresponding
linearized operator about the wave. It is expected that this condition
should in general be analytically verifiable in the case of small
amplitude profiles, but this has so far been shown only on a
case-by-case basis using clever (and difficult to generalize) energy
estimates. Here, we describe how the same set of Evans function tools
that were used to accomplish the original reduction can be used to
show also small-amplitude spectral stability by a direct and readily
generalizable procedure. This approach both recovers the results
obtained by energy methods, and yields new results not previously
obtainable. In particular, we establish one-dimensional stability of
small amplitude relaxation profiles, completing the Evans function
program set out in Mascia&Zumbrun [MZ.1]. Multidimensional stability
of small amplitude viscous profiles will be addressed in a companion
paper [PZ], completing the program of Zumbrun [Z.3].

DCDS

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.## Year of publication

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