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Leonid Berlyand

Publications and source records attributed to Leonid Berlyand.

At least 37 records · Page 2Linked to original sources
arXiv2017

A Two scale $Γ$-convergence Approach for Random Non-Convex Homogenization

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $Γ$-convergence approach and a definition of Young measures on micropatterns which encodes the profiles of the oscillating functions and of functionals. Our abstract result is a lower bound for such energies in terms of a cell problem (on large expanding cells) and the $Γ$-limits of the functionals at the microscale. We show that our method allows to retrieve the results of Dal Maso and Modica in the well-known case of the stochastic homogenization of convex Lagrangians. As an application, we also show how our method allows to stochastically homogenize a variational problem introduced and studied by Alberti and Müller, which is a paradigm of a problem where an additional mesoscale arises naturally due to the non-convexity of the singular perturbation (lower order) terms in the functional.

math.AP↗
arXiv2017

Focusing of Active Particles in a Converging Flow

We consider active particles swimming in a convergent fluid flow in a trapezoid nozzle with no-slip walls. We use mathematical modeling to analyze trajectories of these particles inside the nozzle. By extensive Monte Carlo simulations, we show that trajectories are strongly affected by the background fluid flow and geometry of the nozzle leading to wall accumulation and upstream motion (rheotaxis). In particular, we describe the non-trivial focusing of active rods depending on physical and geometrical parameters. It is also established that the convergent component of the background flow leads to stability of both downstream and upstream swimming at the centerline. The stability of downstream swimming enhances focusing, and the stability of upstream swimming enables rheotaxis in the bulk.

cond-mat.soft↗
arXiv2017

Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.

math.AP↗
arXiv2017

Minimal Model of Directed Cell Motility on Patterned Substrates

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while a those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern and the strength of actin polymerization on the directionality of the crawling cell; these results have applications in motile cell sorting and guiding on engineered substrates.

q-bio.CB↗
arXiv2017

On approximation of Ginzburg-Landau minimizers by $\mathbb S^1$-valued maps in domains with vanishingly small holes

We consider a two-dimensional Ginzburg-Landau problem on an arbitrary domain with a finite number of vanishingly small circular holes. A special choice of scaling relation between the material and geometric parameters (Ginzburg-Landau parameter vs hole radius) is motivated by a recently dsicovered phenomenon of vortex phase separation in superconducting composites. We show that, for each hole, the degrees of minimizers of the Ginzburg-Landau problems in the classes of $\mathbb S^1$-valued and $\mathbb C$-valued maps, respectively, are the same. The presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques.

math.AP↗
arXiv2016

Flagella bending affects macroscopic properties of bacterial suspensions

To survive in harsh conditions, motile bacteria swim in complex environment and respond to the surrounding flow. Here we develop a PDE model describing how the flagella bending affects macroscopic properties of bacterial suspensions. First, we show how the flagella bending contributes to the decrease of the effective viscosity observed in dilute suspension. Our results do not impose tumbling (random re-orientation) as it was done previously to explain the viscosity reduction. Second, we demonstrate a possibility of bacterium escape from the wall entrapment due to the self-induced buckling of flagella. Our results shed light on the role of flexible bacterial flagella in interactions of bacteria with shear flow and walls or obstacles.

math.AP↗
arXiv2016

Phase field model of cell motility: sharp interface limit in sub-critical case

We consider a system of two PDEs arising in modeling of motility of eukariotic cells on substrates. This system consists of the Allen-Cahn equation for the scalar phase field function coupled with another vectorial parabolic equation for the orientation of the actin filament network. The two key properties of this system are (i) presence of gradients in the coupling terms (gradient coupling) and (ii) mass (volume) preservation constraints. We first prove that the sharp interface property of initial conditions is preserved in time. Next we formally derive the equation of the motion of the interface, which is the mean curvature motion perturbed by a nonlinear term that appears due to the properties (i)-(ii). This novel term leads to surprising features of the the motion of the interface. Because of these properties maximum principle and classical comparison techniques do not apply to this system. Furthermore, the system can not be written in a form of gradient flow, which is why recently developed Gamma-convergence techniques also can not be used for the justification of the formal derivation. Such justification is presented in a one-dimensional model problem and it leads to a stability result in a class of 'sharp interface' initial data.

math.AP↗
arXiv2016

Phase-Field Model of Cell Motility: Traveling Waves and Sharp Interface Limit

This letter is concerned with asymptotic analysis of a PDE model for motility of a eukaryotic cell on a substrate. This model was introduced in [1], where it was shown numerically that it successfully reproduces experimentally observed phenomena of cell-motility such as a discontinuous onset of motion and shape oscillations. The model consists of a parabolic PDE for a scalar phase-field function coupled with a vectorial parabolic PDE for the actin filament network (cytoskeleton). We formally derive the sharp interface limit (SIL), which describes the motion of the cell membrane and show that it is a volume preserving curvature driven motion with an additional nonlinear term due to adhesion to the substrate and protrusion by the cytoskeleton. In a 1D model problem we rigorously justify the SIL, and, using numerical simulations, observe some surprising features such as discontinuity of interface velocities and hysteresis. We show that nontrivial traveling wave solutions appear when the key physical parameter exceeds a certain critical value and the potential in the equation for phase field function possesses certain asymmetry.

math.AP↗
arXiv2016

Sharp interface limit in a phase field model of cell motility

We consider a system of two coupled parabolic PDEs introduced in [1] to model motility of eukaryotic cells. We study the asymptotic behavior of solutions in the limit of a small parameter related to the width of the interface in phase field function (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 form of asymptotic expansion is introduced to reduce analysis to a single nonlinear PDE. 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.

math.AP↗
arXiv2015

On an evolution equation in a cell motility model

This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth ($H^2$) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solution exists rely on a maximum principle type argument.

math.AP↗
arXiv2015

Effective Rheological Properties in Semidilute Bacterial Suspensions

Interactions between swimming bacteria have led to remarkable experimentally observable macroscopic properties such as the reduction of the effective viscosity, enhanced mixing, and diffusion. In this work, we study an individual based model for a suspension of interacting point dipoles representing bacteria in order to gain greater insight into the physical mechanisms responsible for the drastic reduction in the effective viscosity. In particular, asymptotic analysis is carried out on the corresponding kinetic equation governing the distribution of bacteria orientations. This allows one to derive an explicit asymptotic formula for the effective viscosity of the bacterial suspension in the limit of bacterium non-sphericity. The results show good qualitative agreement with numerical simulations and previous experimental observations. Finally, we justify our approach by proving existence, uniqueness, and regularity properties for this kinetic PDE model.

math.AP↗
arXiv2014

Complexity reduction in many particles systems with random initial data

We consider the motion of interacting particles governed by a coupled system of ODEs with random initial conditions. Direct computations for such systems are prohibitively expensive due to a very large number of particles and randomness requiring many realizations in their locations in the presence of strong interactions. While there are several approaches that address the above difficulties, none addresses all three simultaneously. Our goal is to develop such a computational approach in order to capture the experimentally observed emergence of correlations in the collective state (patterns due to strong interactions). Our approach is based on the truncation of the BBGKY hierarchy that allows one to go beyond the classical Mean Field limit and capture correlations while drastically reducing the computational complexity. Finally, we provide an example showing a numerical solution of this nonlinear and non-local system.

physics.comp-ph↗
arXiv2013

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator $-\diiv(a\nabla \cdot)$ and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincaré inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

math.NA↗
arXiv2012

Boundary conditions in PDE model of collisions of swimmers

The goal of the paper is to determine boundary conditions in PDE models of collisions of microswimmers in a viscous fluid. We consider two self-propelled spheres (microswimmers) moving towards each other in viscous fluid. We first show that under commonly used no-slip boundary conditions on the fluid-solid interface the microswimmers do not collide which is a generalization of the well-known no-collision paradox for solid bodies (with no self-propulsion) in a viscous fluid. Secondly, we show that the microswimmers do collide when the no-slip boundary conditions are replaced by the Navier boundary conditions which therefore provides an adequate model of microswimmers such as swimming bacteria. The self-propulsion mechanism generates a drag force pulling a bacterium backwards and the collision problem is reduced to the analysis of competition between the drag and self-propulsion. For no-slip this is done by utilizing the Lorentz Reciprocal Theorem and the analytical solution for two solid spheres in the fluid. The analysis for the Navier boundary conditions is based on the variational formulation of Stokes problem. A Poincare type inequality for symmetrized gradient is introduced in this work.

math.AP↗
arXiv2010

Radially symmetric minimizers for a $p$-Ginzburg Landau type energy in $\R^2$

We consider the minimization of a p-Ginzburg-Landau energy functional over the class of radially symmetric functions of degree one. We prove the existence of a unique minimizer in this class, and show that its modulus is monotone increasing and concave. We also study the asymptotic limit of the minimizers as p \rightarrow \infty. Finally, we prove that the radially symmetric solution is locally stable for $p$ in the interval $(2,4]$.

math.AP↗
arXiv2010

Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough ($L^\infty(Ω)$, $Ω\subset \R^d$) coefficients $a(x)$ that, in particular, model media with non-separated scales and high contrast in material properties. We define the flux norm as the $L^2$ norm of the potential part of the fluxes of solutions, which is equivalent to the usual $H^1$-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the {\it transfer property}. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization.

math.AP↗
arXiv2008

Effective Viscosity of Dilute Bacterial Suspensions: A Two-Dimensional Model

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.

physics.bio-ph↗