286 research outputs found
Formal matched asymptotics for degenerate Ricci flow neckpinches
Gu and Zhu have shown that Type-II Ricci flow singularities develop from
nongeneric rotationally symmetric Riemannian metrics on , for all . In this paper, we describe and provide plausibility arguments for a
detailed asymptotic profile and rate of curvature blow-up that we predict such
solutions exhibit
Application of the level-set method to the implicit solvation of nonpolar molecules
A level-set method is developed for numerically capturing the equilibrium
solute-solvent interface that is defined by the recently proposed variational
implicit solvent model (Dzubiella, Swanson, and McCammon, Phys. Rev. Lett. {\bf
104}, 527 (2006) and J. Chem.\Phys. {\bf 124}, 084905 (2006)). In the level-set
method, a possible solute-solvent interface is represented by the zero
level-set (i.e., the zero level surface) of a level-set function and is
eventually evolved into the equilibrium solute-solvent interface. The evolution
law is determined by minimization of a solvation free energy {\it functional}
that couples both the interfacial energy and the van der Waals type
solute-solvent interaction energy. The surface evolution is thus an energy
minimizing process, and the equilibrium solute-solvent interface is an output
of this process. The method is implemented and applied to the solvation of
nonpolar molecules such as two xenon atoms, two parallel paraffin plates,
helical alkane chains, and a single fullerene . The level-set solutions
show good agreement for the solvation energies when compared to available
molecular dynamics simulations. In particular, the method captures solvent
dewetting (nanobubble formation) and quantitatively describes the interaction
in the strongly hydrophobic plate system
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres
Using mixed analytical and numerical methods we investigate the development
of singularities in the heat flow for corotational harmonic maps from the
-dimensional sphere to itself for . By gluing together
shrinking and expanding asymptotically self-similar solutions we construct
global weak solutions which are smooth everywhere except for a sequence of
times at which there occurs the type I blow-up at one
of the poles of the sphere. We show that in the generic case the continuation
beyond blow-up is unique, the topological degree of the map changes by one at
each blow-up time , and eventually the solution comes to rest at the zero
energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
Evolution of elastic thin films with curvature regularization via minimizing movements
Phase Slips and the Eckhaus Instability
We consider the Ginzburg-Landau equation, , with complex amplitude . We first analyze the phenomenon of
phase slips as a consequence of the {\it local} shape of . We next prove a
{\it global} theorem about evolution from an Eckhaus unstable state, all the
way to the limiting stable finite state, for periodic perturbations of Eckhaus
unstable periodic initial data. Equipped with these results, we proceed to
prove the corresponding phenomena for the fourth order Swift-Hohenberg
equation, of which the Ginzburg-Landau equation is the amplitude approximation.
This sheds light on how one should deal with local and global aspects of phase
slips for this and many other similar systems.Comment: 22 pages, Postscript, A
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics
We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for in bounded domains \Om\sub\R^n which arises in game theory. We prove that solutions converge to if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with , i.e. the finite-time blow-up is global
Caproic acid production by chain elongation in mixed culture fermentation from domestic and granular sludge in batch experiments.
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