1,625 research outputs found
Existence and sharp localization in velocity of small-amplitude Boltzmann shocks
Using a weighted -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 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 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 analyse
Dispersive Stabilization
Ill posed linear and nonlinear initial value problems may be stabilized, that
it converted to to well posed initial value problems, by the addition of purely
nonscalar linear dispersive terms. This is a stability analog of the Turing
instability. This idea applies to systems of quasilinear Schr\"odinger
equations from nonlinear optics
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