Rate of convergence to self-similarity for the fragmentation equation in L^1 spaces

In a recent result by the authors (ref. [1]) it was proved that solutions of the self-similar fragmentation equation converge to equilibrium exponentially fast. This was done by showing a spectral gap in weighted $L^2$ spaces of the operator defining the time evolution. In the present work we prove that there is also a spectral gap in weighted $L^1$ spaces, thus extending exponential convergence to a larger set of initial conditions. The main tool is an extension result in ref. [4]

Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

AbstractWe study the asymptotic behavior of linear evolution equations of the type ∂tg=Dg+Lg−λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=−∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=−∂x(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂tf=Lf.By means of entropy–entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part

Deconstructing Interrupts with Ara

The emulation of DHTs is a natural issue. After years of important research into lambda calculus, we demonstrate the refinement of multicast frameworks. In order to answer this challenge, we disprove that despite the fact that spreadsheets and kernels are mostly incompatible, 128 bit architectures and active networks are regularly incompatible

Self-Similarity for Ballistic Aggregation Equation

We consider ballistic aggregation equation for gases in which each particle is iden- ti?ed either by its mass and impulsion or by its sole impulsion. For the constant aggregation rate we prove existence of self-similar solutions as well as convergence to the self-similarity for generic solutions. For some classes of mass and/or impulsion dependent rates we are also able to estimate the large time decay of some moments of generic solutions or to build some new classes of self-similar solutions

Spectral analysis of semigroups and growth-fragmentation equations

The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson-Phillips series in the spirit of those used in [87,82,48,81]. (2) On the other hand, we present the semigroup spectral analysis for three important classes of ''growth-fragmentation" equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick-Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated remarkable profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in \cite{MR2114128,MR2536450} for the cell division model with (almost) constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in \cite{MR2162224,MR2114413} which have been established under similar assumptions to those made in the present work

Re-entry survival analysis and ground risk assessment of space debris considering by-products generation

[EN] Space debris that re-enter the Earth's atmosphere can be partially or fully ablated along the trajectory path after hitting the atmosphere layers, once these become denser (approximately below 82 km). This paper combines reentry survival analysis to by-product generation analyses according to specific trajectory analysis and different levels of modelling within the re-entry simulation tool. Particular attention is made on metallic alloy decomposition and metallic oxides formation from the debris' materials ablation. Generic alloys present within satellite constructions are considered. The flow field in the induced shock layer is considered to be in non-equilibrium and the trajectory tool is based on a 3DOF object-oriented approach. The by-product analyses give important information on emitted species in the atmosphere at different altitudes, and the risk of substances reaching the ground is evaluated as a function of the initial break-up altitude. The non-equilibrium atmospheric chemistry within the shock layer has a significant impact for the re-entry analysis.This work was supported by the Swiss Government Excellence Scholarship (ESKAS No. 2019.0535) awarded by Federal Commission for Scholarships (FCS). The collaboration with UPV was partially financed as part of an activity performed with TAS-I in the context of an ESA subcontract ARA, under ITT-A0/1-8558/16/NL/KML.Park, S.; Navarro-Laboulais, J.; Leyland, P.; Mischler, S. (2021). Re-entry survival analysis and ground risk assessment of space debris considering by-products generation. Acta Astronautica. 179:604-618. https://doi.org/10.1016/j.actaastro.2020.09.03460461817

Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section

This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming $S^{n-1}$ integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel--Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes large data near a local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in [3], by E. Carneiro and the authors, that permits to study such propagation without additional conditions on the collision kernel. Finally, an $L^{p}$-stability result (with $1\leq p\leq\infty$) is presented assuming the aforementioned condition.Comment: 19 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

Galvanically enhanced fretting-crevice corrosion of cemented femoral stems

The Ultima TPS MoM THR was designed and developed as a 2nd generation MoM THR specifically aimed at younger more active patients due to the anticipated low wear rates and increased longevity of MoM THRs. In 2010, published clinical data highlighted the early failure of the Ultima TPS MoM due to fretting-crevice corrosion at the stem-cement interface. Since 2010 similar observations have been reported by other clinical centres implicating competitor products as well as the Ultima TPS MoM THR. In an attempt to replicate the electrochemical reaction and interactions established across MoM THR systems, fretting-crevice corrosion tests subjected to galvanic coupling were conducted. Galvanic coupling was seen to significantly increase the rates of corrosion under static and dynamic conditions. This was due to the large potential differences developed across the system between active and passive areas, increasing the rates of corrosion and metallic ion release from the stem-cement interface