Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics

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The Theory of Composites

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

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The heterogeneous multiscale method

Simulation of the multi-scale convergence in computational homogenization approaches

https://www.math.nyu.edu/faculty/majda/pdfFiles/PhysRep%201999%20Majda%20Kramer.pdf

Simplified models for turbulent diffusion : theory, numerical modelling, and physical phenomena

Homogenization of some problems with gradient constraints

We establish the compactness relative to Γ-convergence for integrands satisfying a coercivity and growth estimate with a variable exponent subject to the so-called logarithmic condition In particular, standard functionals with the same exponents of coercivity and growth are included We prove the complete integral representation for Γ-limit functionals

Gamma-Convergence of Integrands with Nonstandard Coercivity and Growth Conditions

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.

On the Navier-Stokes equations: Existence theorems and energy equalities

In the first part of the paper, we study the stabilization of solutions to nonlinear parabolic equations of the p-Laplace type in L2(ℝn) for 2n/(n + 2) < p ⩽ n. The proof is based on estimates uniform in time in L2(ℝn, |x|s) for p < n. For p = n, a similar estimate holds with |x|s replaced by (ln(2+|x|))s. In the second part of the paper, we prove a uniform stabilization criterion for bounded solutions. This criterion generalizes a widely known result independently proved by Zhikov and Kamin in 1976 for linear parabolic equations.

Stabilization of solutions to nonlinear parabolic equations of the p-Laplace type

Let H be a separable Hilbert space and let H1 be a Hilbert space whose elements are vector functions f(x) (0≤x<∞) with domain H. Define the scalar product in H1 to be (f(x), g(x))1=\(\int_0^\infty {(f(x),g(x))_H dx}\). This paper concerns the study of the negative spectrum of the operatorl(y) =−yπ +Q(x)y, y'(0)−hy(0)=0, y(x) e H1, where Q(x) is a strongly measurable operator function in H which is completely continuous for almost all x, and where h is a fixed completely continuous operator in H. There is 1 reference.

V. V. Zhikov

Papers

Homogenization of some problems with gradient constraints

Gamma-Convergence of Integrands with Nonstandard Coercivity and Growth Conditions

On the Navier-Stokes equations: Existence theorems and energy equalities

Stabilization of solutions to nonlinear parabolic equations of the p-Laplace type

Conditions for the negative spectrum of the Schrodinger equation operator to be discrete and finite