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 $p = \rho^\gamma$ with the constraint that $1< \gamma\le {\frac{5}{3}}$. They asked if a different approach could go beyond the $\gamma > {\frac{5}{3}}$ threshold. We provide an affirmative answer to their question, and prove stability of affine flows and global existence for all $\gamma >1$, thus also establishing global existence for the shallow water equations when $\gamma=2$.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 SL(2,R){\rm SL}(2,{\mathbb R})

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 ${\rm SL}(2,{\mathbb R})$. In the case of perfect fluids, the motion is given by geodesic flow in ${\rm SL}(2,{\mathbb R})$ with the Euclidean metric, while for magnetically conducting fluids (MHD), the motion is governed by a harmonic oscillator in ${\rm SL}(2,{\mathbb R})$. 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 $t\to\pm\infty$. For MHD, solutions are bounded and generically quasi-periodic and recurrent.Comment: 60 pages, 7 figure

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