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The main purpose of this paper is to capture the asymptotic behaviour for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted p-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.
The main objective of this paper is to establish the convergence for the fractional $p$-Laplacian of sequences of nonnegative functions with $p>2$. Furthermore, we show the blow-up phenomena for solutions to the extended Nirenberg problem modelled by fractional $p$-Laplacian with the prescribed negative functions.
Direct numerical simulations of particle-laden turbulent channel flows at friction Reynolds number $Re_\tau$ from $600$ to $2000$ have been performed to examine the near-wall particle streaks. Different from the well-observed small-scale particle streaks in near-wall turbulence of low $Re_\tau$, the present results show large-scale particle streaks through the computational domain formed for relatively high-inertia particles at high $Re_\tau$. Transferred by large-scale sweep and ejection events ($Q^-$), these high-inertia particles preferentially accumulate in near-wall regions beneath the large-scale low-speed flow streaks observed in the logarithmic region. The corresponding Stokes numbers are associated with the lifetime of large-scale $Q^-$ structures, which increases as the Reynolds number grows. The small-scale particle streaks with a typical Stokes number $St_\nu \approx 30$ are mainly driven by the $Q^-$ structures in the buffer layer, whose lifetime is approximately $30$ in viscous time unit. Therefore, we propose a new structure-based Stokes number normalized by the lifetime of $Q^-$ structures of different scales. The relevant flow scales that control the formation of the large-scale particle streaks are parameterized by the structure-based Stokes number. The small-scale (large-scale) particle streaks are most prominent when the buffer-layer (large-scale) structure-based Stokes number approaches unity. The present findings reveal that formation of near-wall particle streaks is governed by the $Q^-$ structures of different scales, and the particles with different inertia respond efficiently to the $Q^-$ structures of corresponding scales with respect to the particle translational motion.
The problem of the magnetic attitude tracking control is studied for a gravity gradient microsatellite in orbital transfer. The contributions of the work are mainly shown in two aspects: (1) the design of an expected attitude trajectory; (2) a method of the magnetic attitude tracking control. In orbital transfer, the gravity gradient microsatellite under a constant thrust shows complicated dynamic behaviours. In order to damp out the pendular motion, the gravity gradient microsatellite is subject to the the attitude tracking problem. An expected attitude trajectory is designed based on dynamic characteristics revealed in the paper, which not only ensures the flight safety of the system, but also reduces the energy consumption of the controller. Besides, the control torque produced by a magnetorquer is constrained to lie in a two-dimensional plane orthogonal to the magnetic field, so an auxiliary compensator is proposed to improve the control performance, which is different from existing magnetic control methods. In addition, a sliding mode control based on the compensator is presented, and the Lyapunov stability analysis is performed to show the global convergence of the tracking error. Finally, a numerical case of the gravity gradient microsatellite is studied to demonstrate the effectiveness of the proposed tracking control.
Gongyi is the birthplace and one of the most famous production areas of white porcelain in China. In this study, white porcelain samples from the Northern Wei to Tang Dynasties excavated from Baihe and Huangye kiln sites were analyzed to investigate microstructure and its physicochemical basis. The result demonstrates that the formation of an interaction layer of Anorthite crystals and the accompanied phase-separation structure at glaze-body boundary is a common character of microstructure. And there is little probability for crystal precipitation within the glaze layer.
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