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In this paper, we are interested in the decay properties of the axisymmetric solutions to the stationary Navier-Stokes equations in R 3 with finite Dirichlet integral. We establish some new a priori decay rates for the second order derivatives of the velocity. The result is obtained by combining the weighted energy estimates, the Brezis-Galleout inequality and the scaling argument.

Self-confidence is one of the most important conditions needed for athletes to succeed. The way in which the self-confidence of athletes may be improved is an eternal topic in training and competition. This study aimed to examine whether unconscious goal priming can improve athletes’ explicit and implicit self-confidence through three experiments. Experiment 1 was aimed at exploring the effect of unconscious goal priming on athletes’ self-confidence. For the purpose of this experiment, 120 athletes were randomly divided into a priming group and a control group. Experiment 2 was aimed at examining the training effect of unconscious goal priming on athletes’ self-confidence and for the purpose of this experiment 16 rhythmic gymnastics athletes were selected as subjects. Experiment 3 was aimed at checking the replication of the training effect of unconscious goal priming on athletes’ self-confidence employing 16 judo athletes as subjects. The results indicated that unconscious goal priming improved athletes’ explicit self-confidence but not implicit self-confidence, while unconscious goal priming intervention improved both athletes’ explicit and implicit self-confidence. These results provide an economical and convenient alternative for improving athletes’ self-confidence.

We show that the Stokes–Darcy system, which governs flows through adjacent porous and pure-fluid domains in the two-domain approach without forced filtration, can be recovered from the Helmholtz minimal dissipation principle. While the continuity of normal velocity across the interface is imposed explicitly for mass conservation, only the Beavers–Joseph–Saffman–Jones (BJSJ) interface boundary condition is imposed implicitly, and the balance of the normal-force interface boundary condition appears naturally in the variational process. This set of interface boundary conditions is well-accepted in the mathematics community. We show that these interfacial boundary conditions, at the physically important small-Darcy-number regime, are consistent with continuity of pressure across the interface condition. The tangential velocity and pressure are discontinuous in general but the discontinuity is of the order of the square root of the Darcy number. Hence these interfacial conditions are all approximately consistent in the physically important small-Darcy-number regime. The leading order dynamics in the pure fluid zone is governed by the Stokes system with the no-slip no-penetration boundary condition on the interface between the free zone and porous media at a small Darcy number. The leading order non-trivial dynamics in porous media is governed by the Darcy equation with the pressure on the interface prescribed by the pressure of the leading order Stokes flow in the pure fluid zone. Such a semi-decoupled approach has long been used by the groundwater community. Our result is the first rigorous work quantifying the error of this intuitive approach and relating different interfacial conditions.

In this paper, we investigate the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a helically symmetric spatial domain. When data is assumed to be helical invariant and satisfies the compatibility condition, we prove this problem has at least one helical invariant solution., Comment: 24 pages