The Hyperbolic Yang–Mills Equation for Connections in an Arbitrary Topological Class

This is the third part of a four-paper sequence, which establishes the Threshold Conjecture and the Soliton-Bubbling versus Scattering Dichotomy for the energy critical hyperbolic Yang–Mills equation in the (4 + 1)-dimensional Minkowski space-time. This paper provides basic tools for considering the dynamics of the hyperbolic Yang–Mills equation in an arbitrary topological class at an optimal regularity. We generalize the standard notion of a topological class of connections on Rd, defined via a pullback to the one-point compactification Sd= Rd∪ { ∞} , to rough connections with curvature in the critical space Ld2(Rd). Moreover, we provide excision and extension techniques for the Yang–Mills constraint (or Gauss) equation, which allow us to efficiently localize Yang–Mills initial data sets. Combined with the results in the previous paper (Oh and Tataru in The hyperbolic Yang–Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions, 2017. arXiv:1709.09332), we obtain local well-posedness of the hyperbolic Yang–Mills equation on R1+d(d≥ 4) in an arbitrary topological class at optimal regularity in the temporal gauge (where finite speed of propagation holds). In addition, in the energy subcritical case d = 3, our techniques provide an alternative proof of the classical finite energy global well-posedness theorem of Klainerman–Machedon (Ann. Math. (2) 142(1):39–119, 1995. https://doi.org/10.2307/2118611), while also removing the smallness assumption in the temporal-gauge local well-posedness theorem of Tao (J. Differ. Equ. 189(2):366–382, 2003. https://doi.org/10.1016/S0022-0396(02)00177-8). Although this paper is a part of a larger sequence, the materials presented in this paper may be of independent and general interest. For this reason, we have organized the paper so that it may be read separately from the sequence

Klein tunneling through an oblique barrier in graphene ribbons

We study a transmission coefficient of graphene nanoribbons with a top gate which acts as an oblique barrier. Using a Green function method based on the Dirac-like equation, scattering among transverse modes due to the oblique barrier is taken into account numerically. In contrast to the 2-dimensional graphene sheet, we find that the pattern of transmission in graphene ribbons depends strongly on the electronic structure in the region of the barrier. Consequently, irregular structures in the transmission coefficient are predicted while perfect transmission is still calculated in the case of metallic graphene independently of angle and length of the oblique barrier

Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and diffusive effects

Analytical observations on the aerodynamics of a delta wing with leading edge flaps

The effect of a leading edge flap on the aerodynamics of a low aspect ratio delta wing is studied analytically. The separated flow field about the wing is represented by a simple vortex model composed of a conical straight vortex sheet and a concentrated vortex. The analysis is carried out in the cross flow plane by mapping the wing trace, by means of the Schwarz-Christoffel transformation into the real axis of the transformed plane. Particular attention is given to the influence of the angle of attack and flap deflection angle on lift and drag forces. Both lift and drag decrease with flap deflection, while the lift-to-drag ratioe increases. A simple coordinate transformation is used to obtain a closed form expression for the lift-to-drag ratio as a function of flap deflection. The main effect of leading edge flap deflection is a partial suppression of the separated flow on the leeside of the wing. Qualitative comparison with experiments is presented, showing agreement in the general trends

Theoretical studies on flapped delta wings

The effects of leading edge flaps on the aerodynamic characteristics of a low aspect-ratio delta wing are studied theoretically. As an extension of the classical crossflow plane analysis and in order to include separated shear layers, an analogy between three dimensional steady conical and two dimensional unsteady self-similar flows is explored. This analogy provides a simple steady-unsteady relationship. The criteria for the validity of the steady-unsteady analogy are also examined. Two different theoretical techniques are used to represent the separated shear layers based on the steady-unsteady analogy, neglecting the trailing edge effect. In the first approach, each vortex system is represented by a pair of concentrated vortices connected to the separation points by straight feeding sheets. In the second approach, the vortex cloud method is adopted for simulating the flow field in the crossflow plane. The separated shear layers are replaced with a cloud of discrete vortices and the boundary element method is employed to represent the wing trace by a vorticity distribution. A simple merging scheme is used to model the core region of the vortical flow as a single vortex by imposing a restriction on the shear layer rotation angle. The results are compared with experiments and with results from 3-D panel calculations

Influence of external disturbances and compressibility on free turbulent mixing

It is shown that disturbances in external flow can significantly affect, by as much as an order of magnitude, the turbulent mixing rate in free shear layers and that the length scale of the external flow disturbances is as important as the amplitude. The difference between the effect of wide-band and narrow-band disturbances is stressed. The model for pressure fluctuation term in the kinetic energy equation is included in a two-equation model. The reduced spreading rate in high Mach number, high Reynolds number, adiabatic, free turbulent shear layers is predicted

Betweenness centrality correlation in social networks

Scale-free (SF) networks exhibiting a power-law degree distribution can be grouped into the assortative, dissortative and neutral networks according to the behavior of the degree-degree correlation coefficient. Here we investigate the betweenness centrality (BC) correlation for each type of SF networks. While the BC-BC correlation coefficients behave similarly to the degree-degree correlation coefficients for the dissortative and neutral networks, the BC correlation is nontrivial for the assortative ones found mainly in social networks. The mean BC of neighbors of a vertex with BC $g_i$ is almost independent of $g_i$, implying that each person is surrounded by almost the same influential environments of people no matter how influential the person is.Comment: 4 pages, 4 figures, 1 tabl