On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

Solutions of a Lagrangian system on

A Lagrangian system on

Phase separation in aqueous solutions of lens gamma-crystallins: special role of gamma s.

We have studied liquid-liquid phase separation in aqueous ternary solutions of calf lens gamma-crystallin proteins. Specifically, we have examined two ternary systems containing gamma s--namely, gamma IVa with gamma s in water and gamma II with gamma s in water. For each system, the phase-separation temperatures (Tph (phi)) alpha as a function of the overall protein volume fraction phi at various fixed compositions alpha (the "cloud-point curves") were measured. For the gamma IVa, gamma s, and water ternary solution, a binodal curve composed of pairs of coexisting points, (phi I, alpha 1) and (phi II, alpha II), at a fixed temperature (20 degrees C) was also determined. We observe that on the cloud-point curve the critical point is at a higher volume fraction than the maximum phase-separation temperature point. We also find that typically the difference in composition between the coexisting phases is at least as significant as the difference in volume fraction. We show that the asymmetric shape of the cloud-point curve is a consequence of this significant composition difference. Our observation that the phase-separation temperature of the mixtures in the high volume fraction region is strongly suppressed suggests that gamma s-crystallin may play an important role in maintaining the transparency of the lens.