The paper is a comprehensive study of the $L_p$ and the Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients in the half space and cylindrical domains with conormal derivative boundary condition. For t
he $L_p$ estimates, we assume that the leading coefficients are only bounded measurable in the $t$ variable and $VMO$ with respect to $x$. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the $x$ variable; the coefficients are Holder continuous in both variables.
Initially defined by the IAU in 1958, the galactic coordinate system was thereafter in 1984 transformed from the B1950.0 FK4-based system to the J2000.0 FK5-based system. In 1994, the IAU recommended that the dynamical reference system FK5 be replace
d by the ICRS, which is a kinematical non-rotating system defined by a set of remote radio sources. However the definition of the galactic coordinate system was not updated. We consider that the present galactic coordinates may be problematic due to the unrigorous transformation method from the FK4 to the FK5, and due to the non-inertiality of the FK5 system with respect to the ICRS. This has led to some confusions in applications of the galactic coordinates. We tried to find the transformation matrix in the framework of the ICRS after carefully investigating the definition of the galactic coordinate system and transformation procedures, however we could not find a satisfactory galactic coordinate system that is connected steadily to the ICRS. To avoid unnecessary misunderstandings, we suggest to re-consider the definition of the galactic coordinate system which should be directly connected with the ICRS for high precise observation at micro-arcsecond level.
We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as an indepen
dent variable (M2), and the two-nearby-orbits method without projection operations and with proper time as an independent variable (M3). An analysis indicates that M1 and M3 do not need any projection operation. In general, the values of LEs from the three methods are almost the same. As an advantage, M3 is simpler to use than M2. In addition, we propose to construct the invariant fast Lyapunov indictor (FLI) with two-nearby-trajectories and give its algorithm in order to quickly distinguish chaos from order. Taking a static axisymmetric spacetime as a physical model, we apply the invariant FLIs to explore the global dynamics of phase space of the system where regions of chaos and order are clearlyidentified.
In this paper, we will give an extension of Moks theorem on the generalized Frankel conjecture under the condition of the orthogonal bisectional curvature.
