Fluid dynamic limit to compressible Euler equations from compressible
Navier-Stokes equations and Boltzmann equation has been an active topic with
limited success so far. In this paper, we consider the case when the solution
of the Euler equations is a Riemann solution consisting two rarefaction waves
and a contact discontinuity and prove this limit for both Navier-Stokes
equations and the Boltzmann equation when the viscosity, heat conductivity
coefficients and the Knudsen number tend to zero respectively.

The hybrid linear modeling problem is to identify a set of d-dimensional
affine sets in a D-dimensional Euclidean space. It arises, for example, in
object tracking and structure from motion. The hybrid linear model can be
considered as the second simplest (behind linear) manifold model of data. In
this paper we will present a very simple geometric method for hybrid linear
modeling based on selecting a set of local best fit flats that minimize a
global l1 error measure.

In this paper, we study the asymptotic stability of rarefaction waves for the
compressible isentropic Navier-Stokes equations with density-dependent
viscosity. First, a weak solution around a rarefaction wave to the Cauchy
problem is constructed by approximating the system and regularizing the initial
values which may contain vacuum state. Then some global in time estimates on
the weak solution are obtained. Based on these uniform estimates, the vacuum
states are shown to vanish in finite time and the weak solution we constructed
becomes a unique strong one.