Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the non-existence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties.Comment: 20 pages, 4 figures, To appear in: SIAM Journal of Mathematical Analysis, 202

Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on a flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions by using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus

Validity of single-channel model for a spin-orbit coupled atomic Fermi gas near Feshbach resonances

We theoretically investigate a Rashba spin-orbit coupled Fermi gas near Feshbach resonances, by using mean-field theory and a two-channel model that takes into account explicitly Feshbach molecules in the close channel. In the absence of spin-orbit coupling, when the channel coupling $g$ between the closed and open channels is strong, it is widely accepted that the two-channel model is equivalent to a single-channel model that excludes Feshbach molecules. This is the so-called broad resonance limit, which is well-satisfied by ultracold atomic Fermi gases of $^{6}$Li atoms and $^{40}$K atoms in current experiments. Here, with Rashba spin-orbit coupling we find that the condition for equivalence becomes much more stringent. As a result, the single-channel model may already be insufficient to describe properly an atomic Fermi gas of $^{40}$K atoms at a moderate spin-orbit coupling. We determine a characteristic channel coupling strength $g_{c}$ as a function of the spin-orbit coupling strength, above which the single-channel and two-channel models are approximately equivalent. We also find that for narrow resonance with small channel coupling, the pairing gap and molecular fraction is strongly suppressed by SO coupling. Our results can be readily tested in $^{40}$K atoms by using optical molecular spectroscopy.Comment: 6 pages, 6 figure

Facile Synthesis and Versatilities of Polyanthraquinoylamine Nanofibril Bundles with Self Stability and High Carbon Yield

A facile synthesis for nanosized conducting polymers with inherent self-stability and multi-functionalities is a main challenge. Here we simply synthesize intrinsically self-stabilized nanofibril bundles of poly(1-anthraquinoylamine) (PAQ) by a template-free method. The critical polymerization parameters were studied to significantly optimize the synthesis, size, properties, and functionalities of the resulted fine nanofibrils with a diameter of ca. 30 nm and length of ~6 μm. The PAQ obtained with ammonium persulfate possesses higher polymerization yield, purer composition, higher conductivity, better melting behaviour, higher thermostability, lower burning enthalpy, and slower degradation than that with other oxidants. Furthermore, the polymer nanofibrils exhibit high self-stability, powerful redispersibility, high purity, and clean surface because of a complete avoidance of the contamination from external stabilizer. The PAQ exhibits widely controllable conductivity moving across ten orders of magnitudes from 10^-9^ to 50 S/cm, photoluminescence, lead-ion adsorbability, very high thermostability in air and extremely high char yield in nitrogen at 1000˚C. These materials would be useful as advanced materials including photoluminescent materials, highly cost-effective carbon precursors, sorbents of toxic metal ions, and cost-efficient conductive nanocomposite with low percolation threshold