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

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

Estimates for the Boltzmann collision operator via radial symmetry and Fourier transform

We extend the $L^p$-theory of the Boltzmann collision operator by using classical techniques based in the Carleman representation and Fourier analysis, allied to new ideas that exploit the radial symmetry of this operator. We are then able to greatly simplify existent technical proofs in this theory, extend the range, and obtain explicit sharp constants in some convolution-like inequalities for the gain part of the Boltzmann collision operator.Comment: 14 page

The Galaxy Kinematics from the Cepheids with the Proper Motions from the GAIA DR1 Catalog

The sample of classic Cepheids with known distances and line-of-sight velocities is supplemented by the proper motions from the Gaia DR1 catalog. From spatial velocities of 260 stars the components of the peculiar Solar velocity: (U,V,W)_\odot=(7.90,11.73,7.39)+/-(0.65,0.77,0.62) km/s, parameters of the Galactic rotation curve: \Omega_0 =28.840+/-.33 km/s/kpc, \Omega'_0=-4.05+/-0.10 km/s/kpc^2, \Omega''_0=0.805+/-0.067 km/s/kpc^3 are obtained. For the adopted Galactocentric Solar distance R_0=8 kpc the linear circular velocity of the Local Standard of Rest is found as V_0=231+/-6 km/s.Comment: 11 pages, 4 figures, 2 tables. Accepted for publication in Astronomy Letters, 2017, Issue