A blowup criterion for ideal viscoelastic flow

We establish an analog of the Beale-Kato-Majda criterion for singularities of smooth solutions of the system of PDE arising in the Oldroyd model for ideal viscoelastic flow

Short Gamma-Ray Bursts with Extended Emission Observed with Swift/BAT and Fermi/GBM

Some short GRBs are followed by longer extended emission, lasting anywhere from ~10 to ~100 s. These short GRBs with extended emission (EE) can possess observational characteristics of both short and long GRBs (as represented by GRB 060614), and the traditional classification based on the observed duration places some of them in the long GRB class. While GRBs with EE pose a challenge to the compact binary merger scenario, they may therefore provide an important link between short and long duration events. To identify the population of GRBs with EE regardless of their initial classifications, we performed a systematic search of short GRBs with EE using all available data (up to February 2013) of both Swift/BAT and Fermi/GBM. The search identified 16 BAT and 14 GBM detected GRBs with EE, several of which are common events observed with both detectors. We investigated their spectral and temporal properties for both the spikes and the EE, and examined correlations among these parameters. Here we present the results of the systematic search as well as the properties of the identified events. Finally, their properties are also compared with short GRBs with EE observed with BATSE, identified through our previous search effort. We found several strong correlations among parameters, especially when all of the samples were combined. Based on our results, a possible progenitor scenario of two-component jet is discussed.Comment: Published in MNRAS; matched to the published versio

The Two Dimensional Liquid Crystal Droplet Problem with Tangential Boundary Condition

This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove the boundary of the droplet is a chord-arc curve with its normal vector field in the VMO space, and its arc-length parametrization belongs to the Sobolev space $H^{3/2}$. In fact, the boundary curves of such droplets closely resemble the so-called Weil-Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is also studied

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation-strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under the smallness assumptions only on the size of initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.Comment: Different contributions of strain and rotation of the deformation are studied for viscoelastic fluids of Oldroyd-B type in 2