24 research outputs found
Global existence of near-affine solutions to the compressible Euler equations
We establish global existence of solutions to the compressible Euler
equations, in the case that a finite volume of ideal gas expands into vacuum.
Vacuum states can occur with either smooth or singular sound speed, the latter
corresponding to the so-called physical vacuum singularity when the enthalpy
vanishes on the vacuum wave front like the distance function. In this instance,
the Euler equations lose hyperbolicity and form a degenerate system of
conservation laws, for which a local existence theory has only recently been
developed. Sideris found a class of expanding finite degree-of-freedom
global-in-time affine solutions, obtained by solving nonlinear ODEs. In three
space dimensions, the stability of these affine solutions, and hence global
existence of solutions, was established by Had\v{z}i\'{c} \& Jang with the
pressure-density relation with the constraint that . They asked if a different approach could go beyond
the threshold. We provide an affirmative answer to
their question, and prove stability of affine flows and global existence for
all , thus also establishing global existence for the shallow water
equations when .Comment: 51 pages, details added to Section 4.7, to appear in Arch. Rational
Mech. Ana
Affine motion of 2d incompressible fluids surrounded by vacuum and flows in
The affine motion of two-dimensional (2d) incompressible fluids surrounded by
vacuum can be reduced to a completely integrable and globally solvable
Hamiltonian system of ordinary differential equations for the deformation
gradient in . In the case of perfect fluids, the
motion is given by geodesic flow in with the
Euclidean metric, while for magnetically conducting fluids (MHD), the motion is
governed by a harmonic oscillator in . A complete
classification of the dynamics is given including rigid motions, rotating
eddies with stable and unstable manifolds, and solutions with vanishing
pressure. For perfect fluids, the displacement generically becomes unbounded,
as . For MHD, solutions are bounded and generically
quasi-periodic and recurrent.Comment: 60 pages, 7 figure
Establishment of Intein-Mediated Protein Ligation under Denaturing Conditions: C-Terminal Labeling of a Single-Chain Antibody for Biochip Screening
A single ion-conducting polymer electrolyte based on PEO and anionic nanoparticles: electrical and NMR characterization
We report on electrical properties of fluorine-functionalized nanoscopicTiO2and its reaction with molten Li, which produces a single ion Li conductor. It is believed that the fluorines and the lithiums are concentrated in a 2-4 nm shell around the TiO2 core, based on XRD and vibrational spectroscopic evidence.Mixing this material with PEO results in a polymer electrolyte with favorable mechanical as well as electrical properties. For example, a sample containing ~10 wt% PEO demonstrates an ionic conductivity of 1.5 x 10-5 S/cm at 20oC and 1.0 x 10-4 S/cm at 80oC, with a Li+ transference number of unity. Additional characterization, including electrochemical stability window, exchange current density, and variable temperature broadband electrical spectroscopy (covering a frequency range of 10-2 to 107 Hz) Multinuclear (1H, 7Li and 19F) solid state NMR measurements, including spectra, relaxation, and transport as a function of PEO/TiO2:F,Li ratio will be presented