Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off

In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a moderate angular singularity). We prove a result of existence and uniqueness of solutions in a close-to-equilibrium regime for this equation in weighted Sobolev spaces with a polynomial weight, contrary to previous works on the subject, all developed with a weight prescribed by the equilibrium. It is the first result in this more physically relevant frameworkfor this equation. Moreover, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay of the linearized equation. Let us highlight the fact that a key point of the development of our Cauchy theory is the proof of new regularization estimates in short time for the linearized operator thanks to pseudo-differential tools.Comment: arXiv admin note: text overlap with arXiv:1709.0994

Second order mean field games with degenerate diffusion and local coupling

We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation

Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields

In the present study we consider planar piecewise linear vector fields with two zones separated by the straight line $x=0$. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has center, a real one for $y<0$ and a virtual one for $y>0$, and such that the real center is a global center. Then, working with a first order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.Comment: 24 pages, 7 figure

Black anodic coatings for space applications: study of the process parameters, characteristics and mechanical properties

Black inorganic anodized aluminium alloys are used for managing passive thermal control on spacecraft and for avoiding stray light in optical equipment. Spalling of these coatings has sometimes been observed after thermal cycling on 2XXX and 7XXX aluminium alloys. This phenomenon could generate particulate contamination in satellites and may affect mission lifetime. In this work, the influences of the four main steps of the process (pretreatments, sulphuric anodizing, colouring and sealing) on the coating characteristics have been studied for a 7175 T7351 aluminium alloy. The chemical heterogeneity of the coating has been underlined, and its mechanical behaviour observed through crazing. Scratch-testing, used to evaluate coating adhesion to its substrate, revealed the negative impact of thermal cycling