Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Modelling and asymptotic analysis of the concentration difference in a nanoregion between an influx andoutflux diffusion acrossnarrow windows

When a flux of Brownian particles is injected in a narrow window located on the surface of a bounded domain, these particles diffuse and can eventually escape through a cluster of narrow windows. At steady state, we compute asymptotically the distribution of concentration between the different windows. The solution is obtained by solving Laplace’s equation using Green’s function techniques and second-order asymptotic analysis, and depends on the influx amplitude, the diffusion properties, as well as the geometrical organization of all the windows, such as their distances and the mean curvature. We explore the range of validity of the present asymptotic expansions using numerical simulations of the mixed boundary value problem. Finally, we introduce a length scale to estimate how deep inside a domain a local diffusion current can spread.

Structure and Properties of a Local Diffusion Discrete Coating Applied to High Speed Steel

Investigations of metallographic and mechanical properties of local diffusion discrete oxide coating on high-speed steel Р6М5 have been carried out. It was found that the technology of discrete oxidation makes it possible to increase the hardness by 31% in relation to the uncoated material, and the wear resistance of the cutting tool with oxidation is 1.5-3 times higher than that of the tool hardened by the standard ion-plasma technology.

Microscopic kinetics of isothermal sintering of Fe-20 % (mаs.) Mo alloy

Purpose of work. To investigate the features of microscopic kinetics of peritectoid transformation in Fe-Mo system alloys in an isothermal mode. Experimental part. Microscopic analysis of samples on light (Jenaphot 2000, K. Zeiss) and scanning electron (REM 106I, Selmi) microscopes, X-ray spectral microanalysis of the component’s concentrations distribution between the phases, X-ray phase analysis (Rigaku Ultima IV diffractometer). Results. Microstructure changes, phase composition and crystal lattices parameters of the phase constituents of the powder alloy during sintering at 920 °C were investigated. Variation in the phase constituents mass fraction during 7 hours of the isothermal exposure is analyzed. The formation of anomalous diffusion porosity at the beginning of the process, the nonmonotonic change in the phase constituents fraction and formation of intermediate phases with an unstable component’s concentration are the main features of the microscopic kinetics. The sintering mechanism is proposed. Scientific novelty. A local peritectoid transformation existence at the Fe/Mo interface was established by analyzing the local diffusion flows of components atoms. This transformation occurs upon isothermal supply of Mo atoms with the formation of a cooperative peritectoid structural constituents according to the α- Fe + Mo → α + μ scheme with residual Mo crystals. Formulation of the problem. This work aims to clarify the phenomenological theory of peritectoid transformation during isothermal α-Fe grains enrichment with molybdenum by studying the features of microscopic kinetics in the Fe-Mo system alloys. Practical value. Peritectoid (α + μ) with branched phase соnstituents of cooperative genesis forms a developed system of local diffusion flows of Mo atoms in α -Fe. This increases the molybdenum peritectoid transformation rate at a relatively low sintering temperature for these alloys and reduces the energy consumption in the technological process.

Two-scale, non-local diffusion in homogenised heterogeneous media

We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods.

Transport of neutral and charged nanorods across varying-section channels

We study the dynamics of neutral and charged rods embedded in varying-section channels. By means of systematic approximations, we derive the dependence of the local diffusion coefficient on both the...