We consider the 3D Euler equations with Coriolis force (EC) in the whole space. We show long-time solvability in Besov spaces for high speed of rotation Ω and arbitrary initial data. For that, we obtain Ω-uniform estimates and a blow-up criterion of BKM type in our framework. Our initial data class is larger than previous ones considered for (EC) and covers borderline cases of the regularity. The uniqueness of solutions is also discussed.

Vertical jump performance is often assessed using jump-and-reach tests. The exact procedure used for determining standing reach height and jump height has a large effect on the resultant displacement. The purpose of this investigation was to determine the influence of 4 methods of standing reach height measurement and Vertec jump height measurement against 2 force plate methods of jump displacement determination (impulse and flight-time methods). Fifteen men with various training backgrounds performed 2 each of countermovement, restricted (no arm swing) and static start vertical jumps. Reach height was determined using 4 methods; either a 1- or overlapped 2-hand reach, flat footed or with plantar flexion. All jumps were performed on a force platform. The best jump of each type based on Vertec displacement was used for analysis. Repeated-measures of analysis of variance for each jump type was used for analysis with Bonferroni post hoc for pairwise comparisons of jump measurement style. All jump displacements for similar types were significantly intercorrelated with a minimum r-value of 0.84. Impulse vs. flight time was the only pairwise comparison of measurement type for which similar values were noted. The one-hand reach with plantar flexion was the method of reach that was closest to the impulse and flight-time methods, and thus should be the preferred choice when using jump-and-reach tests to determine jump displacement. In all cases, the Vertec overestimates the displacement of the COM based on force plate methods. When comparing groups of individuals from different data sets, one must consider both the method of reach height (if performed) and jump displacement to make valid comparisons. If plantar flexion with a 1-hand reach is not used during reach measurement, jump displacement will be erroneously high.

In this paper we show the eventual local positivity property for higher-order heat equations (including noninteger order). As a consequence, we give a positive answer for an open problem stated by Barbatis and Gazzola [Contemp. Math. 594 (2013)] for polyharmonic heat equations. Moreover, we obtain some polynomial decay properties of solutions.

The surface goniometry protocol described herein appeared to be reliable for relatively lean young men and women. Although measures were precise to 1.0°, it appears a difference of 3° may be needed to detect a real difference in Q-angles when measured in this fashion.

This work proves the convergence in L 1 (R 2 ) towards an Oseen vortex-like solution to the dissipative quasi-geostrophic equations for several sets of initial data with suitable decay at infinity. The relative entropy method applies in a direct way for solving this question in the case of signed initial data and the difficulty lies in showing the existence of unique global solutions for the class of initial data for which all properties needed in the entropy approach are met. However, the estimates obtained for the constructed global solutions in L 1 ∩ L 2 show the asymptotic simplification of the solutions even for unsigned initial data emphasizing the character of this equation to behave linearly for large times.2000 Mathematics Subject Classification: 35Q, 35B40. Key words and phrases: quasi-geostrophic equations, asymptotics of solutions, self-similarity. The paper is in final form and no version of it will be published elsewhere.[95] 96 J. A. CARRILLO AND L. C. F. FERREIRAwith κ > 0 and γ ∈ [0, 1]. The velocity field u = (u 1 , u 2 ) is divergence free, ∇ · u = 0, and determined from the potential temperature θ through the stream function ψIn other words, the velocity field is given in terms of Riesz transforms of the potential temperature that will be denoted by the operator u = R[θ].Let us remark the existence of a special self-similar solution of the 2DQG. Fix κ = 1 for the remainder of the paper and let γ ∈and the fact that Fourier transform preserves radial symmetry. Using the expression of the velocity field (1.2) and the fact that both ψ G γ and G γ are radial functions, we have the following orthogonality propertyis the fundamental solution of the operator ∂ ∂t − ∆ γ , then it takes the delta Dirac distribution δ 0 as initial data weakly as measures. In other words, θ(t, x) = G γ (t, x) with its corresponding stream function is a source-type solution for the 2DQG equations.We will show that this particular self-similar solution of 2DQG for γ = 1, i.e. the heat kernel G = G 1 for the equationplays the same role as the Oseen vortex for the two dimensional Navier-Stokes (2DNSV) equations in vorticity-velocity formulation. The Oseen vortex has recently been proved by T. Gallay and C.E. Wayne [15,16] to be globally asymptotically stable for all initial integrable vorticities regardless of their sign. Moreover, they obtain decay rates towards the Oseen vortex under suitable additional conditions on the initial data. The space L 1 (R 2 ) is natural for the 2DNSV due to the homogeneity of the Biot-Savart law while for the 2DQG the right homogeneity space is L 2 (R 2 ). However, the "mass" is preserved, at least formally, for (1.3). Existence of self-similar solutions with local basins of attraction has been shown in [4] in Lorentz spaces and some spaces of tempered distributions with the right homogeneity, i.e. θ 0 ∈ L 2,∞ (R 2 ). Assuming that the initial data belongs to the strong L p space with the right homogeneity θ 0 ∈ L 2 (R 2 ), it was proved in [4] that there exist global solutions for small initia...