We consider the stationary problem of the Navier-Stokes equations in R n for n ≥ 3. We show existence, uniqueness and regularity of solutions in the homogeneous Besov spacė B −1+ n p p,q which is the scaling invariant one. As a corollary of our results, a self-similar solution is obtained. For the proof, several bilinear estimates are established. The essential tool is based on the paraproduct formula and the imbedding theorem in homogeneous Besov spaces. up to the present. For instance, Chen [6] improved regularity of the solution of (NS) in L n , and Secchi [20] treated the solution in L n ∩ L p with p > n.Among these results, it seems important to find more general spaces where the solution of (NS) is obtained. For that purpose, we shall introduce homogeneous Besov spaces. As the first step, it is necessary to seek a suitable function space to handle (NS). To this end, the scaling argument is quite useful. Indeed, it is easy to show that if {u, π, f } satisfies (NS), then so does the family {u λ , π λ , f λ } for all λ > 0, where u λ (x) = λu(λx), π λ (x) = λ 2 π(λx) and f λ (x) = λ 3 f (λx). We call the spaces X and Y with the norms ∥ · ∥ X and ∥ · ∥ Y scaling invariant for the velocity u and the external force f with respect to (NS) if it holds that ∥u λ ∥ X = ∥u∥ X and ∥f λ ∥ Y = ∥f ∥ Y for all λ > 0, respectively. For instance, in the usual Lebesgue space L p , L n (R n ) and L n 3 (R n ) are scaling invariant for u and f with respect to (NS) since it holds that ∥u λ ∥ L n = ∥u∥ L n andhomogeneous Besov spaces enable us to treat more general spaces than Lebesgue ones. In comparison with the Lebesgue spaces, the advantage of Besov spaces to solve the non-stationary problems in R 3 has been established by Cannone [3] and Cannone-Planchon [5]. Indeed, they proved the existence theorem for the Cauchy problem with the initial data u 0 ∈Ḃ −1+ n p p,∞ with n < p < ∞ which is larger than L n . More abstract result including [3] and [5] was shown by Cannone-Meyer [4], which made fully use of the Littlewood-Paley decomposition. Kato [13] and Kozono-Yamazaki [16] obtained the corresponding results to the Morrey space. The purpose of the present paper is to prove existence, uniqueness and regularity of the solutions of (NS) in the homogeneous Besov spaces with the scaling invariant norms. More precisely, if the external force f is sufficiently small inḂ −3+ n p p,q with 1 ≤ p < n, 1 ≤ q ≤ ∞, then there exists a unique solution u ∈Ḃ −1+ n p p,q of (NS). Since such homogeneous Besov spaces for f and u include homogeneous functions with degrees −3 and −1, respectively, our existence theorem necessarily yields the self-similar solution of (NS). The advantage of making use of homogeneous Besov spaces stems from the fact that the scaling argument works, while it is not the case in inhomogeneous Besov spaces. On account of such a scaling property as that of usual L p -spaces, we are successful to construct the stationary solution u ∈Ḃ −1+ n p p,q for small f ∈Ḃ −3+ n p p,q. On the other hand, there is also disadvantage of ho...
Polarimetry is a powerful tool for astrophysical observations that has yet to be exploited in the X-ray band. For satellite-borne and sounding rocket experiments, we have developed a photoelectric gas polarimeter to measure X-ray polarization in the 2-10 keV range utilizing a time projection chamber (TPC) and advanced micro-pattern gas electron multiplier (GEM) techniques. We carried out performance verification of a flight equivalent unit (1/4 model) which was planned to be launched on the NASA Gravity and Extreme Magnetism Small Explorer (GEMS) satellite. The test was performed at Brookhaven National Laboratory, National Synchrotron Light Source (NSLS) facility in April 2013. The polarimeter was irradiated with linearly-polarized monochromatic X-rays between 2.3 and 10.0 keV and scanned with a collimated beam at 5 different detector positions. After a systematic investigation of the detector response, a modulation factor ≥35% above 4 keV was obtained with the expected polarization angle. At energies below 4 keV where the photoelectron track becomes short, diffusion in the region between the GEM and readout strips leaves an asymmetric photoelectron image. A correction method retrieves an expected modulation angle, and the expected modulation factor, ∼20% at 2.7 keV. Folding the measured values of modulation through an instrument model gives sensitivity, parameterized by minimum detectable polarization (MDP), nearly identical to that assumed at the preliminary design review (PDR).
The scientific objective of the X-ray Advanced Concepts Testbed (XACT) is to measure the X-ray polarization properties of the Crab Nebula, the Crab pulsar, and the accreting binary Her X-1. Polarimetry is a powerful tool for astrophysical investigation that has yet to be exploited in the X-ray band, where it promises unique insights into neutron stars, black holes, and other extreme-physics environments. With powerful new enabling technologies, XACT will demonstrate X-ray polarimetry as a practical and flight-ready astronomical technique. Additional technologies that XACT will bring to flight readiness will also provide new X-ray optics and calibration capabilities for NASA missions that pursue space-based X-ray spectroscopy, timing, and photometry.
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