On the ill-posedness of the Prandtl equations in three space dimensions

In this paper, we give an instability criterion for the Prandtl equations in three space variables, which shows that the monotonicity condition of tangential velocity fields is not sufficient for the well-posedness of the three dimensional Prandtl equations, in contrast to the classical well-posedness theory of the Prandtl equations in two space variables under the Oleinik monotonicity assumption of the tangential velocity. Both of linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linear stable for the three dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work cite{LWY} on the well-posedness theory for the three dimensional Prandtl equations with special structure.

A Fan Beam Model for Radio Pulsars. I. Observational Evidence

We propose a novel beam model for radio pulsars based on the scenario that the broadband and coherent emission from secondary relativistic particles, as they move along a flux tube in a dipolar magnetic field, forms a radially extended sub-beam with unique properties. The whole radio beam may consist of several sub-beams, forming a fan-shaped pattern. When only one or a few flux tubes are active, the fan beam becomes very patchy. This model differs essentially from the conal beam models in the respects of beam structure and predictions on the relationship between pulse width and impact angle $beta$ (the angle between line of sight and magnetic pole) and the relationship between emission intensity and beam angular radius. The evidence for this model comes from the observed patchy beams of precessional binary pulsars and three statistical relationships found for a sample of 64 pulsars, of which $beta$ were mostly constrained by fitting polarization position angle data with the Rotation Vector Model. With appropriate assumptions, the fan beam model can reproduce the relationship between 10% peak pulse width and $|beta|$, the anticorrelation between the emission intensity and $|beta|$, and the upper boundary line in the scatter plot of $|beta|$ versus pulsar distance. An extremely patchy beam model with the assumption of narrowband emission from one or a few flux tubes is studied and found unlikely to be a general model. The implications of the fan beam model to the studies on radio and gamma-ray pulsar populations and radio polarization are discussed.

Growth dynamics and thickness-dependent electronic structure of topological insulator Bi2Te3 thin films on Si

We use real-time reflection high energy electron diffraction intensity oscillation to establish the Te-rich growth dynamics of topological insulator thin films of Bi2Te3 on Si(111) substrate by molecular beam epitaxy. In situ angle resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy and ex situ transport measurements reveal that the as-grown Bi2Te3 films without any doping are an intrinsic topological insulator with its Fermi level intersecting only the metallic surface states. Experimentally, we find that the single-Dirac-cone surface state develops at a thickness of two quintuple layers (2 QL). Theoretically, we show that the interaction between the surface states from both sides of the film, which is determined by the penetration depth of the topological surface state wavefunctions, sets this lower thickness limit.