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Oleksandr Misiats

Publications and source records attributed to Oleksandr Misiats.

15 recordsLinked to original sources
arXiv2026

On the Optimal Control Problem of Stochastic Semilinear Partial Differential Equations with Non-Globally Lipschitz Coefficients

In this paper, we study optimal control problems for stochastic semilinear partial differential equations, which lack the maximum principle, and whose coefficients do not have bounded Frechet derivatives. We propose an approximation scheme for the corresponding optimization problem, and prove convergence of the approximating solutions on both finite and infinite time intervals.

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arXiv2026

Long-time behavior of a nonlocal and non-monotone SPDE-ODE system arising in electrophysiology

This paper concerns a coupled semilinear SPDE-ODE system modelling the electropermeabilization phenomenon, which designates a transient increase in cell membrane permeability induced by short, high-voltage electric pulses. We present a stochastically perturbed electroporation model that couples electrostatic equations for the electric potential in the extra- and intracellular domains and a nonlinear evolution law for the transmembrane potential jump with an ordinary differential equation describing the porosity degree of the membrane. We prove the existence and uniqueness of a variational solution of the resulting coupled stochastic PDE-ODE system. Its long-time behavior is governed by the corresponding invariant measure for which we establish the regularity of its support. The ergodicity of this invariant measure is further established for a truncated nonlinear reaction term, corresponding to the case of a bounded electric potential. The main technical challenge arises from the nonlinear reaction term, which is neither Lipschitz continuous nor locally monotone. We also present a numerical example, computing the solution and its time averages for both additive and multiplicative noise, that provides an indication for the existence of an invariant measure.

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arXiv2026

Long Time Behavior of Stochastic Thin Film Equation

We consider the stochastic thin-film equation with linear deterministic and stochastic Itô perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t \to \infty$ is investigated. It is shown that in square mean the $L^\infty$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.

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arXiv2023

Long Time Behavior of Stochastic Thin Film Equation

In this paper we consider a stochastic thin-film equation with a one dimensional Gaussian Stratonovych noise. We establish the existence of non-negative global weak martingale solution, and study its long time asymptotic properties. In particular, we show the solution almost surely converges to the average value of the initial condition. Furthermore, using the regularized equations and adapted entropy functionals, we establish the exponential asymptotic decay of the solution in the uniform norm.

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arXiv2023

Thin Film Equations with Nonlinear Deterministic and Stochastic Perturbations

In this paper we consider stochastic thin-film equation with nonlinear drift terms, colored Gaussian Stratonovych noise, as well as nonlinear colored Wiener noise. By means of Trotter-Kato-type decomposition into deterministic and stochastic parts, we couple both of these dynamics via a discrete-in-time scheme, and establish its convergence to a non-negative weak martingale solution.

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arXiv2022

An energy minimization approach to twinning with variable volume fraction

In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example {involving} a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call ``twinning with variable volume fraction.'' Two 1996 papers by Chopra et. al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. The upper bounds are ansatz-based, providing full details about some (nearly) optimal microstructures. The lower bounds are ansatz-free, so they explain why no other arrangement of the two phases could be significantly better.

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arXiv2021

On global existence and blowup of solutions of stochastic Keller-Segel type equation

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.

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arXiv2021

Strong solutions and asymptotic behavior of bidomain equations with random noise

In this paper we study the conditions for the existence of strong solutions (both local and global) for stochastic bidomain equations. To this end, we use apriori energy estimates and Serrin-type theorems. We further address the asymptotic behavior of the solutions, which includes the analysis of small stochastic perturbations and large deviations. In a separate section we specify the support of the invariant measure.

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arXiv2020

Invariant Measure for Stochastic Functional Differential Equations in Hilbert Spaces

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.

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arXiv2020

On minimizers of an anisotropic liquid drop model

We consider a variant of Gamow's liquid drop model with an anisotropic surface energy. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. We show that for smooth anisotropies, in the small nonlocality regime, minimizers converge to the Wulff shape in $C^1$-norm and quantify the rate of convergence. We also obtain a quantitative extension of the energy of any minimizer around the energy of a Wulff shape yielding a geometric stability result. For certain crystalline surface tensions we can determine the global minimizer and obtain its exact energy expansion in terms of the nonlocality parameter.

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arXiv2019

Singular perturbation of an elastic energy with a singular weight

We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by Alberti and Müller in 2001 we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on functions of slope $\pm 1$ and of period depending on the location in the domain and the weights in the energy.

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arXiv2014

Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains

In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + σ(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $σ$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions $f$ and $σ$ and elliptic operators $A$ for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $A$ is the Shrödinger-type operator $A = \frac{1}ρ(\text{div} ρ\nabla u)$ where $ρ= e^{-|x|^2}$ is the Gaussian weight.

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arXiv2014

Convergence of Space-Time Discrete Threshold Dynamics to Anisotropic Motion by Mean Curvature

We analyze the continuum limit of a thresholding algorithm for motion by mean curvature of one dimensional interfaces in various space-time discrete regimes. The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn equation which is a typical model for the motion of materials phase boundaries. Our results extend the existing statements which are applicable mostly in semi-discrete (continuous in space and discrete in time) settings. The motivations of this work are twofolds: to investigate the interaction between multiple small parameters in nonlinear singularly perturbed problems, and to understand the anisotropy in curvature for interfaces in spatially discrete environments. In the current work, the small parameters are the the spatial and temporal discretization step sizes $\triangle x = h$ and $\triangle t = τ$. We have identified the limiting description of the interfacial velocity in the (i) sub-critical ($h \ll τ$), (ii) critical ($h = O(τ)$), and (iii) super-critical ($h \gg τ$) regimes. The first case gives the classical isotropic motion by mean curvature, while the second produces intricate pinning and de-pinning phenomena and anisotropy in the velocity function of the interface. The last case produces no motion (complete pinning).

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arXiv2011

Ginzburg-Landau model with small pinning domains

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, $\v$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $Ω\subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\d Ø$, with topological degree ${\rm deg}_{\d Ø} (g) = d >0$. Our main result is that, for small $\v$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to 1. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by {\it local renormalized energy} which does not depend on the external boundary conditions.

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