The up-scaling is a passage from a description of processes at the microscopic scale to a description at the macroscopic scale. The multiscale analysis of heterogeneous structures and, in particular, the up-scaling technique is actually an important and effective tool of mathematical modeling in physics, technique and biology. The mathematical theory on the up-scaling (passage from the micro-scale to the macro-scale) called homogenization appeared in early seventies of the XXth century independently in the USSR, France, Italy and USA. Although since then the homogenization theory has been considerably developed and generalized, there are still open problems in constructing realistic models of new materials and engineering structures, biological processes, especially combining different scales or discrete and continuous approaches.
The special issue contains the best selected papers of the Fifth Conference on Multiscale Methods and Modeling devoted to the up-scaling in engineering and medicine (3MUEM-2015 conference www.mccme.bmstu.ru). This conference brought together the well-known specialists in the topic and young researchers and students.
The multiscale methods combine microscopic and macroscopic descriptions of the phenomena and are usually based on asymptotic and numerical analysis of the microscopic model equations. The application of these methods allows constructing new materials with given properties, creating new more adequate and more precise models in biology and medicine. The theoretical results on the mathematical analysis of multiscale models are presented together with the numerical and computer experiments.
The first part of the special issue contains theoretical results on the spectral homogenization and resolvent analysis for elliptic operators and the boundary homogenization. Applications to biological and medical problems are considered in the second part. Applications to some engineering problems are considered in the third part.
In the paper by Yu.M. Meshkova and T.A. Suslina a generalized resolvent approximation is constructed for an elliptic second order matrix-valued operator with rapidly oscillating periodic coefficients. An error estimate is proved for this approximation in terms of two parameters: the homogenization small parameter that is the ratio of the scales and the spectral parameter
. The resolvent analysis for high order elliptic operators with rapidly oscillating coefficients is developed in the paper by S. Pastukhova, where the exact estimates for the error of asymptotic approximations are given. Elliptic operators considered in these papers describe, in particular, the diffusion-convection steady processes.
A three-dimensional model for the interaction of a thin stratified rigid plate and a viscous fluid layer is considered in the article by I. Malakhova-Ziablova, G. Panasenko, and R. Stavre. This problem depends on a small parameter which is the ratio of the thickness of the plate and that of the fluid layer. The Young modulus of the plate is high: it is of order
. The variational analysis of this model (including the existence, uniqueness of the solution and its regularity) is provided and an asymptotic expansion of the solution is constructed and justified. The limit problem contains a non-standard boundary condition for the Stokes equations. The existence, uniqueness and regularity of its solution are proved. The asymptotic analysis is applied to the partial asymptotic dimension reduction of the solid phase and the derivation of the asymptotically exact junction conditions between two-dimensional and three-dimensional models of the plate. This analysis helps to choose an appropriate boundary condition for hemodynamics in a blood vessel. The paper by G.Panasenko and V.Volpert is devoted to the diffusion equation with discrete absorption described by a sum of Dirac
-functions. Their supports are located at the nodes of some regular grid with the distance between them determined by the integral of solution. This model describes contraction of biological tissue when cells consume some substance influencing their interaction. In the one-dimensional formulation the existence of solutions of the discrete problem and their convergence to the solution of the limiting homogenized problem are proved.
A model for the spreading of a substance through an incompressible fluid in a perforated domain is considered in the paper by A. Brillard, D. Gómez, M. Lobo, E. Pérez and T.A. Shaposhnikova. The fluid flows in a domain containing a periodic set of perforations
placed along an inner surface
. The size of the perforations is much smaller than the size of the characteristic period
and a strong nonlinear sorption takes place at the boundary of perforations. This problem is homogenized by means of the energy method and a nonlinear strange term appears in the homogenized model.
The paper by E.M. Zilonova and A.S. Bratus deals with an important applied problem of the optimization the delivery of an antibiotic to a bacteria population consisting of two communicating compartments, susceptible and resistant to the antibiotic. Another paper with strong biological applications is presented by V.S. Rozova and A.S. Bratus. It considers a mathematical model of interactions between cancer cells and immune system. The model consisting of five ordinary differential equation describes the behavior of a tumour cell population, populations of immune cells as well as kinetics of the chemotherapy drug concentration in the bloodstream. In order to find the best possible pattern of the drug administration the optimal control analysis is provided.
An eigenvalue problem for a perturbed two-dimensional resonance oscillator is considered in the paper by A.V.Pereskokov where the excitation potential is given by a Hartree-type nonlinearity with a smooth self-action potential. An asymptotic analysis of eigenvalues and eigenfunctions near the lower boundaries of spectral clusters based on the quantum averaging is developed. The Cauchy problem for the second order Petrovsky-parabolic system of equations with the Dini-continuous coefficients is considered in the paper by E.A. Baderko and M.F. Cherepova, the uniqueness of a solution is proved in the Tikhonovs class. A boundary value problem for the radiation transfer equation with reflection and refraction conditions describing a stationary process of the radiation transfer in multilayered semitransparent for radiation medium, consisting of several parallel vertical layers is considered in the paper by A. Amosov. The mathematical analysis of this problem is provided: the existence and uniqueness of a solution to this problem with general data spaces are proved, a priori estimates for the solution are given.
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