Strong Convergence towards homogeneous cooling states for dissipative Maxwell models

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogenous cooling state present in the literature implies the strong convergence in Sobolev norms and in the $L^1$ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the $L^1$ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.Comment: 2 figure

The System of Molecular Clouds in the Gould Belt

Based on high-latitude molecular clouds with highly accurate distance estimates taken from the literature, we have redetermined the parameters of their spatial orientation. This system can be approximated by a 350x235x140 pc ellipsoid inclined by the angle i=17+/-2 degrees to the Galactic plane with the longitude of the ascending node l{\Omega}=337+/-1 degrees. Based on the radial velocities of the clouds, we have found their group velocity relative to the Sun to be (u0,v0,w0) = (10.6,18.2,6.8)+/-(0.9,1.7,1.5) km/s. The trajectory of the center of the molecular cloud system in the past in a time interval of ~60 Myr has been constructed. Using data on masers associated with low-mass protostars, we have calculated the space velocities of the molecular complexes in Orion, Taurus, Perseus, and Ophiuchus. Their motion in the past is shown to be not random.Comment: 15 pages, 4 figure

Estimation of the SolarGalactocentric Distance and Rotation Velocity from Near-Solar-Circle Objects

We have tested the method of determining the solar Galactocentric distance R_0 and Galactic rotation velocity $V_0$ modified by Sofue et al. using near-solar-circle objects. The motion of objects relative to the local standard of rest has been properly taken into account. We show that when such young objects as star-forming regions or Cepheids are analyzed, allowance for the perturbations produced by the Galactic spiral density wave improves the statistical significance of the estimates. The estimate of R_0=7.25\pm0.32 kpc has been obtained from 19 star-forming regions. The following estimates have been obtained from a sample of 14 Cepheids (with pulsation periods P>5^d): R_0=7.66\pm0.36 kpc and V_0=267\pm 17 km s^{-1}. We consider the influence of the adopted Oort constant A and the character of stellar proper motions (Hipparcos or UCAC4). The following estimates have been obtained from a sample of 18 Cepheids with stellar proper motions from the UCAC4 catalog: R_0=7.64\pm0.32 kpc and V_0=217\pm 11 km s^{-1}.Comment: 13 pages, 2 figures, 3 table

Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails

We consider the Boltzmann equations for mixtures ofMaxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.Comment: 20 page

Tanaka Theorem for Inelastic Maxwell Models

We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance

Galactic Rotation Curve and Spiral Density Wave Parameters from 73 Masers

Based on kinematic data on masers with known trigonometric parallaxes and measurements of the velocities of HI clouds at tangential points in the inner Galaxy, we have refined the parameters of the Allen-Santillan model Galactic potential and constructed the Galactic rotation curve in a wide range of Galactocentric distances, from 0 to 20 kpc. The circular rotation velocity of the Sun for the adopted Galactocentric distance $R_0=8$ kpc is $V_0=239\pm16$ km s$^{-1}$. We have obtained the series of residual tangential, $\Delta V_{\theta}$, and radial, $\Delta V_R,$ velocities for 73 masers. Based on these series, we have determined the parameters of the Galactic spiral density wave satisfying the linear Lin.Shu model using the method of periodogram analysis that we proposed previously. The tangential and radial perturbation amplitudes are $f_\theta=7.0\pm1.2$ km s$^{-1}$ and $f_R=7.8\pm0.7$ km s$^{-1}$, respectively, the perturbation wavelength is $\lambda=2.3\pm0.4$ kpc, and the pitch angle of the spiral pattern in a two-armed model is $i=-5.2^\circ\pm0.7^\circ$. The phase of the Sun $\chi_\odot$ in the spiral density wave is $-50^\circ\pm15^\circ$ and $-160^\circ\pm15^\circ$ from the residual tangential and radial velocities, respectively.Comment: 14 pages, 3 figures, 2 table

Orientation Parameters of the Cepheid System in the Galaxy

Based on the distribution of long-period Cepheids, we have redetermined the orientation parameters of their principal plane in the Galaxy. Based on 299 Cepheids with heliocentric distances $r<20$ kpc and pulsation periods $P\geq5^d$, we have found the directions of the three principal axes of the position ellipsoid: $L_1=281.0\pm0.1^\circ,$ $B_1=-1.9\pm0.1^\circ,$ $L_2= 11.0\pm0.7^\circ,$ $B_2=0.2\pm0.1^\circ$ and $L_3=275.9\pm0.7^\circ,$ $B_3=88.1\pm0.1^\circ$. Thus, the line of nodes $l_\Omega=L_3+90^\circ=5.9^\circ$ is very close to the direction to the Galactic center; the Cepheid symmetry plane is inclined to the Galactic plane approximately by $-2^\circ$ in the direction of the first axis ($L_1$). The direction of the line of nodes found from old Cepheids ($P<5^d$) differs significantly and is $l_\Omega=298^\circ$. The elevation of the Sun above the Galactic plane has been estimated from 365 closer stars ($r<4$ kpc) without any constraint on the pulsation period to be $h_\odot=23\pm5$ pc.Comment: 10 pages, 3 figure