758 research outputs found
Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion
Phase field models frequently provide insight to phase transitions, and are
robust numerical tools to solve free boundary problems corresponding to the
motion of interfaces. A body of prior literature suggests that interface motion
via surface diffusion is the long-time, sharp interface limit of microscopic
phase field models such as the Cahn-Hilliard equation with a degenerate
mobility function. Contrary to this conventional wisdom, we show that the
long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free
energy undergoes coarsening, reflecting the presence of bulk diffusion, rather
than pure surface diffusion. This reveals an important limitation of phase
field models that are frequently used to model surface diffusion
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case
We develop a one--parameter family of hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear elliptic equations in divergence-form in a bounded Lipschitz domain. Using Brouwer's Fixed Point Theorem, we show existence and uniqueness of the solution. In addition, we derive an error bound in a broken energy norm which is optimal in h and mildly suboptimal in p
Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers
We show the existence of global-in-time weak solutions to a general class of
coupled FENE-type bead-spring chain models that arise from the kinetic theory
of dilute solutions of polymeric liquids with noninteracting polymer chains.
The class of models involves the unsteady incompressible Navier-Stokes
equations in a bounded domain in two or three space dimensions for the velocity
and the pressure of the fluid, with an elastic extra-stress tensor appearing on
the right-hand side in the momentum equation. The extra-stress tensor stems
from the random movement of the polymer chains and is defined by the Kramers
expression through the associated probability density function that satisfies a
Fokker-Planck-type parabolic equation, a crucial feature of which is the
presence of a center-of-mass diffusion term. We require no structural
assumptions on the drag term in the Fokker-Planck equation; in particular, the
drag term need not be corotational. With a square-integrable and
divergence-free initial velocity datum for the Navier-Stokes equation and a
nonnegative initial probability density function for the Fokker-Planck
equation, which has finite relative entropy with respect to the Maxwellian of
the model, we prove the existence of a global-in-time weak solution to the
coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the
absence of a body force, the weak solution decays exponentially in time to the
equilibrium solution, at a rate that is independent of the choice of the
initial datum and of the centre-of-mass diffusion coefficient.Comment: 75 page
The Polar Regions of Cassiopeia A: The Aftermath of a Gamma Ray Burst?
Probably not, but it is interesting nevertheless to investigate just how
close Cas A might have come to generating such an event. Focusing on the
northeast jet filaments, we analyze the polar regions of the recently acquired
very deep 1 Ms Chandra X-ray observation. We infer that the so-called "jet"
regions are indeed due to jets emanating from the explosion center, and not due
to polar cavities in the circumstellar medium at the time of explosion. We
place limits on the equivalent isotropic explosion energy in the polar regions
(around 2.3 x 10^52 ergs), and the opening angle of the x-ray emitting ejecta
(around 7 degrees), which give a total energy in the NE jet of order 10^50
ergs; an order of magnitude or more lower than inferred for "typical" GRBs.
While the Cas A progenitor and explosion exhibit many of the features
associated with GRB hosts, e.g. extensive presupernova mass loss and rotation,
and jets associated with the explosion, we speculate that the recoil of the
compact central object, with velocity 330 km/s, may have rendered the jet
unstable. In such cases the jet rapidly becomes baryon loaded, if not truncated
altogether. Although unlikely to have produced a gamma ray burst, the jets in
Cas A suggest that such outflows may be common features of core-collapse SNe.Comment: 35 pages, 7 figures, accepted by Ap
A simple and optimal ancestry labeling scheme for trees
We present a ancestry labeling scheme for trees. The
problem was first presented by Kannan et al. [STOC 88'] along with a simple solution. Motivated by applications to XML files, the label size was
improved incrementally over the course of more than 20 years by a series of
papers. The last, due to Fraigniaud and Korman [STOC 10'], presented an
asymptotically optimal labeling scheme using
non-trivial tree-decomposition techniques. By providing a framework
generalizing interval based labeling schemes, we obtain a simple, yet
asymptotically optimal solution to the problem. Furthermore, our labeling
scheme is attained by a small modification of the original solution.Comment: 12 pages, 1 figure. To appear at ICALP'1
Numerical analysis of a topology optimization problem for Stokes flow
T. Borrvall and J. Petersson [Topology optimization of fluids in Stokes flow,
International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107]
developed the first model for topology optimization of fluids in Stokes flow.
They proved the existence of minimizers in the infinite-dimensional setting and
showed that a suitably chosen finite element method will converge in a weak(-*)
sense to an unspecified solution. In this work, we prove novel regularity
results and extend their numerical analysis. In particular, given an isolated
local minimizer to the analytical problem, we show that there exists a sequence
of finite element solutions, satisfying necessary first-order optimality
conditions, that strongly converges to it. We also provide the first numerical
investigation into convergence rates
Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers
We show the existence of global-in-time weak solutions to a general class of
coupled Hookean-type bead-spring chain models that arise from the kinetic
theory of dilute solutions of polymeric liquids with noninteracting polymer
chains. The class of models involves the unsteady incompressible Navier-Stokes
equations in a bounded domain in two or three space dimensions for the velocity
and the pressure of the fluid, with an elastic extra-stress tensor appearing on
the right-hand side in the momentum equation. The extra-stress tensor stems
from the random movement of the polymer chains and is defined by the Kramers
expression through the associated probability density function that satisfies a
Fokker-Planck-type parabolic equation, a crucial feature of which is the
presence of a center-of-mass diffusion term. We require no structural
assumptions on the drag term in the Fokker-Planck equation; in particular, the
drag term need not be corotational. With a square-integrable and
divergence-free initial velocity datum for the Navier-Stokes equation and a
nonnegative initial probability density function for the Fokker-Planck
equation, which has finite relative entropy with respect to the Maxwellian of
the model, we prove the existence of a global-in-time weak solution to the
coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the
absence of a body force, the weak solution decays exponentially in time to the
equilibrium solution, at a rate that is independent of the choice of the
initial datum and of the centre-of-mass diffusion coefficient.Comment: 86 page
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