Two-parameter homogenization for a Ginzburg-Landau problem in a perforated domain
Leonid Berlyand Petru Mironescu
Let $A$ be an annular type domain in $\mathbb{R}^2$. Let $A_\delta$ be a perforated domain obtained by punching periodic holes of size $\delta$ in $A$; here, $\delta$ is sufficiently small. Suppose that $\J$ is the class of complex-valued maps in $A_\delta$, of modulus $1$ on $\partial A_\delta$ and of degrees $1$ on the components of $\partial A$, respectively $0$ on the boundaries of the holes.

We consider the existence of a minimizer of the Ginzburg-Landau energy

$E_\lambda(u)=\frac 1\2_[\int_{A_\delta}](|\nabla u|^2+\frac\lambda 2(1-|u|^2)^2)$
among all maps in $u\in\J$.

It turns out that, under appropriate assumptions on $\lambda=\lambda(\delta)$, existence is governed by the asymptotic behavior of the $H^1$-capacity of $A_\delta$. When the limit of the capacities is $>\pi$, we show that minimizers exist and that they are, when $\delta\to 0$, equivalent to minimizers of the same problem in the subclass of $\J$ formed by the $\mathbb{S}^1$-valued maps. This result parallels the one obtained, for a fixed domain, in [3], and reduces homogenization of the Ginzburg-Landau functional to the one of harmonic maps, already known from [2].

When the limit is $<\pi$, we prove that, for small $\delta$, the minimum is not attained, and that minimizing sequences develop vortices. In the case of a fixed domain, this was proved in [1].
keywords: perforated domain Ginzburg-Landau energy vortices. homogenization
Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes
Leonid Berlyand Volodymyr Rybalko
We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the $\Gamma$-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes.
keywords: vortices. pinning Ginzburg-Landau functional Homogenization
Asymptotic analysis of an array of closely spaced absolutely conductive inclusions
Leonid Berlyand Giuseppe Cardone Yuliya Gorb Gregory Panasenko
We consider the conductivity problem in an array structure with square closely spaced absolutely conductive inclusions of the high concentration, i.e. the concentration of inclusions is assumed to be close to 1. The problem depends on two small parameters: $\varepsilon$, the ratio of the period of the micro-structure to the characteristic macroscopic size, and $\delta$, the ratio of the thickness of the strips of the array structure and the period of the micro-structure. The complete asymptotic expansion of the solution to problem is constructed and justified as both $\varepsilon$ and $\delta$ tend to zero. This asymptotic expansion is uniform with respect to $\varepsilon$ and $\delta$ in the area $\{\varepsilon=O(\delta^{\alpha}),~\delta =O(\varepsilon^{\beta})\}$ for any positive $\alpha, \beta.$
keywords: two small parameters asymptotic expansion array structure boundary layer. Homogenization
Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque
B. M. Haines Igor S. Aranson Leonid Berlyand Dmitry A. Karpeev
We present a PDE model for dilute suspensions of swimming bacteria in a three-dimensional Stokesian fluid. This model is used to calculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background flow. A bacterium is modeled as an impenetrable prolate spheroid with self-propulsion provided by a point force, which appears in the model as an inhomogeneous delta function in the PDE. The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation). Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background flows, such as a pure straining or planar shear flow, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experiments on suspensions of Bacillus subtilis.
keywords: effective viscosity homogenization. Bacterial suspensions
Leonid Berlyand V. V. Zhikov
This special issue contains selected papers on Homogenization Theory and related topics. It is dedicated to Eugene Khruslov on the occasion of his seventieth birthday. Professor Khruslov made pioneering contributions into this field.
Homogenization problems were first studied in the late nineteenth century (Poisson, Maxwell, Rayleigh) and early twentieth century (Einstein). These studies were based on deep physical intuition allowing these outstanding physicists to solve several specific important problems such as calculating the effective conductivity of a two-phase conductor and the effective viscosity of suspensions. It was not until the early 1960s that homogenization began to gain a rigorous mathematical footing which enabled it to be applied to a wide variety of problem in physics and mechanics. A number of mathematical tools such as the asymptotic analysis of PDEs, variational bounds, heterogeneous multiscale method, and the probabilistic techniques of averaging were developed. Although this theory is a well-established area of mathematics, many fascinating problems remain open. Interesting examples of such problems can be found in the papers of this issue.

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keywords: xxxx
Sharp interface limit in a phase field model of cell motility
Leonid Berlyand Mykhailo Potomkin Volodymyr Rybalko

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic behavior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlinear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parameter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

keywords: Allen-Cahn equation phase field model sharp interface limit cell motility traveling wave
A network model of geometrically constrained deformations of granular materials
K. A. Ariyawansa Leonid Berlyand Alexander Panchenko
We study quasi-static deformation of dense granular packings. In the reference configuration, a granular material is under confining stress (pre-stress). Then the packing is deformed by imposing external boundary conditions, which model engineering experiments such as shear and compression. The deformation is assumed to preserve the local structure of neighbors for each particle, which is a realistic assumption for highly compacted packings driven by small boundary displacements. We propose a two-dimensional network model of such deformations. The model takes into account elastic interparticle interactions and incorporates geometric impenetrability constraints. The effects of friction are neglected. In our model, a granular packing is represented by a spring-lattice network, whereby the particle centers correspond to vertices of the network, and interparticle contacts correspond to the edges. We work with general network geometries: periodicity is not assumed. For the springs, we use a quadratic elastic energy function. Combined with the linearized impenetrability constraints, this function provides a regularization of the hard-sphere potential for small displacements.
   When the network deforms, each spring either preserves its length (this corresponds to a solid-like contact), or expands (this represents a broken contact). Our goal is to study distribution of solid-like contacts in the energy-minimizing configuration. We prove that under certain geometric conditions on the network, there are at least two non-stretched springs attached to each node, which means that every particle has at least two solid-like contacts. The result implies that a particle cannot loose contact with all of its neighbors. This eliminates micro-avalanches as a mechanism for structural weakening in small shear deformation.
keywords: discrete variational inequalities. geometric rigidity granular materials constrained optimization
Renormalized Ginzburg-Landau energy and location of near boundary vortices
Leonid Berlyand Volodymyr Rybalko Nung Kwan Yip
We consider the location of near boundary vortices which arise in the study of minimizing sequences of Ginzburg-Landau functional with degree boundary condition. As the problem is not well-posed --- minimizers do not exist, we consider a regularized problem which corresponds physically to the presence of a superconducting layer at the boundary. The study of this formulation in which minimizers now do exist, is linked to the analysis of a version of renormalized energy. As the layer width decreases to zero, we show that the vortices of any minimizer converge to a point of the boundary with maximum curvature. This appears to be the first such result for complex-valued Ginzburg-Landau type problems.
keywords: near boundary vortices. Ginzburg-Landau energy renormalized energy

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