arXiv subjects

Georgy Kitavtsev

Publications and source records attributed to Georgy Kitavtsev.

13 recordsLinked to original sources

arXiv2024

Composite solutions to a liquid bilayer model

This article continues the research initiated in [17]. We derive explicit formulae for the leading order profiles of eleven types of stationary solutions to a one-dimensional two-layer thin-film liquid model considered with an intermolecular potential depending on both layer heights. The found solutions are composed of the repeated elementary blocks (bulk, contact line and ultra-thin film ones) being consistently asymptotically matched together. We show that once considered on a finite interval with Neumann boundary conditions these stationary solutions are either dynamically stable or weakly translationally unstable. Other composite solutions are found to be numerically unstable and rather exhibit complex coarsening dynamics.

math.AP↗

arXiv2022

Asymptotic decay towards steady states of solutions to very fast and singular diffusion equations

We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having nonhomogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic $H^1$-convergence of the solutions to a unique nonhomogeneous positive steady state generally to hold.

math.AP↗

arXiv2018

Long-time behaviour of solutions to a singular heat equation with an application to hydrodynamics

In this paper, we extend the results of [1] by proving exponential asymptotic $H^1$-convergence of solutions to a one-dimensional singular heat equation with $L^2$-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.

math.AP↗

arXiv2017

Thin film models for active gels

In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamcs in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Lesli-Erickson-Parodi theory. Analytic expression are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.

math.AP↗

arXiv2017

A Compactness and Structure Result for a Discrete Multi-Well Problem with $SO(n)$ Symmetry in Arbitrary Dimension

In this note we combine the "spin-argument" from [KLR15] and the $n$-dimensional incompatible, one-well rigidity result from [LL16], in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an "internal" microstructure on smaller scales.

math.AP↗

arXiv2017

Thermal rupture of a free liquid sheet

We consider a free liquid sheet, taking into account the dependence of surface tension on temperature, or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi one-dimensional thickness, velocity, and temperature profiles in the pinch region in terms of similarity solutions, which posses a universal structure. Our analytical description agrees quantitatively with numerical simulations.

physics.flu-dyn↗

arXiv2016

Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation

We consider the two-dimensional Landau-de Gennes energy with several elastic constants, subject to general $k$-radially symmetric boundary conditions. We show that for generic elastic constants the critical points consistent with the symmetry of the boundary conditions exist only in the case $k=2$. In this case we identify three types of radial profiles: with two, three of full five components and numerically investigate their minimality and stability depending on suitable parameters. We also numerically study the stability properties of the critical points of the Landau-de Gennes energy and capture the intricate dependence of various qualitative features of these solutions on the elastic constants and the physical regimes of the liquid crystal system.

math.AP↗

arXiv2015

Surface Energies Emerging in a Microscopic, Two-Dimensional Two-Well Problem

In this article we are interested in the microscopic modeling of a two-dimensional two-well problem which arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyze the Hamiltonian which was introduced in \cite{KLR14}. It turns out that this class of Hamiltonians allows for a direct control of the discrete second order gradients and for a one-sided comparison with a two-dimensonal spin system. Using this and relying on the ideas of Conti and Schweizer \cite{CS06}, \cite{CS06a}, \cite{CS06c}, which were developed for a continuous analogue of the model under consideration, we derive a (first order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

math.AP↗

arXiv2014

Surface Energies Arising in Microscopic Modeling of Martensitic Transformations

In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By constraining the deformed configurations to special 1D atomic chains with position-dependent elongation vectors for the vertical direction, we show that the structure of ground states under appropriate boundary conditions is close to the macroscopically expected twinned configurations with additional boundary layers localized near the twinning interfaces. In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizing sequences and rigorously derive the corresponding limiting form of the surface energy.

math-ph↗

arXiv2012

Weak solutions to lubrication systems describing the evolution of bilayer thin films

The existence of global nonnegative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

math.AP↗

arXiv2012

Stationary solutions of liquid two-layer thin film models

We investigate stationary solutions of a thin-film model for liquid two-layer flows in an energetic formulation that is motivated by its gradient flow structure. The goal is to achieve a rigorous understanding of the contact-angle conditions for such two-layer systems. We pursue this by investigating a corresponding energy that favors the upper liquid to dewet from the lower liquid substrate, leaving behind a layer of thickness $h_*$. After proving existence of stationary solutions for the resulting system of thin-film equations we focus on the limit $h_*\to 0$ via matched asymptotic analysis. This yields a corresponding sharp-interface model and a matched asymptotic solution that includes logarithmic switch-back terms. We compare this with results obtained using $Γ$-convergence, where we establish existence and uniqueness of energetic minimizers in that limit.

math.AP↗

arXiv2012

Coarsening rates for the dynamics of slipping droplets

We derive reduced finite dimensional ODE models starting from one dimensional lubrication equations describing coarsening dynamics of droplets in nanometric polymer film interacting on a hydrophobically coated solid substrate in the presence of large slippage at the liquid/solid interface. In the limiting case of infinite slip length corresponding in applications to free films a collision/absorption model then arises and is solved explicitly. The exact coarsening law is derived for it analytically and confirmed numerically. Existence of a threshold for the decay of initial distributions of droplet distances at infinity at which the coarsening rates switch from algebraic to exponential ones is shown.

math.AP↗

arXiv2010

Weak solutions to lubrication equations in the presence of strong slippage

The existence of global weak solutions is proved for one-dimensional lubrication models that describe the dewetting process of nanoscopic thin polymer films on hydrophobyzed substrates and take account of large slippage at the polymer-substrate interface. The convergence of these solutions as either the Reynolds number or the capillarity goes to zero, as well as their limiting behaviour as the slip length goes to zero or infinity are investigated.

math.AP↗