We prove an abstract result of existence of "good" generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem $ -\Delta u = a |\nabla u|^p+ b|u|^q+f

We present a model for the life cycle of a dinoflagellate in order to describe blooms. We prove the mathematical well-posedness of the model and the possibility of extinction in finite time of the alga form meaning that the full population is under the cysts from.

Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many

Within this paper, we consider a heterogeneous catalysis system consisting of a bulk phase $ \Omega $ (chemical reactor) and an active surface $ \Sigma = \partial \Omega $ (catalytic surface), between which chemical substances are exchanged via adsorption (transport of mass from the bulk boundary layer adjacent to the surface, leading to surface-accumulation by a transformation into an adsorbed form)

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation $ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $

In this paper, we proved existence and nonexistence of traveling wave solution for a diffusive simple epidemic model with a free boundary in the case where the diffusion coefficient $ d $ of susceptible population is zero and the basic reproduction number is greater than 1. We obtained a curve in the parameter plane which is the boundary between the regions of existence and nonexistence of traveling

In this paper, the evolution of a polygonal spiral curve by the crystalline curvature flow with a pinned center is considered from two viewpoints; a discrete model consisting of an ODE system describing facet lengths and another using level set method. We investigate the difference of these models numerically by calculating the area of an interposed region by their spiral curves. The area difference

We consider the blow-up problems of the power type of stochastic differential equation, $ dX = \alpha X^p(t)dt+X^q(t)dW(t) $. It has been known that there exists a critical exponent such that if $ p $ is greater than the critical exponent then the solution $ X(t) $ blows up almost surely in the finite time. In our research, focus on this critical exponent, we propose a numerical scheme by adaptive

This article deals with a model of signal propagation in excitable media based on a system of reaction-diffusion equations. Such media have the ability to exhibit a large response in reaction to a small deviation from the rest state. An example of such media is the nerve tissue or the heart tissue. The first part of the article briefly describes the origin and the propagation of the cardiac action

The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot class. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy value is. In this article, we discuss two topics on the O'Hara energies. First, we slightly generalize the O'Hara energies and consider a characterization of its

The finite volume method (FVM) as a numerical method can be straightforwardly applied for global as well as local gravity field modelling. However, to obtain precise numerical solutions it requires very refined discretization which leads to large-scale parallel computations. To optimize such computations, we present a special class of numerical techniques that are based on a physical decomposition

We consider a boundary value problem for the stationary Stokes problem and the corresponding pressure-Poisson equation. We propose a new formulation for the pressure-Poisson problem with an appropriate additional boundary condition. We establish error estimates between solutions to the Stokes problem and the pressure-Poisson problem in terms of the additional boundary condition. As boundary conditions

In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.

Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the

Automatic detection of objects in photos and images is beneficial in various scientific and industrial fields. This contribution suggests an algorithm for segmentation of color images by the means of the parametric mean curvature flow equation and CIE94 color distance function. The parametric approach is enriched by the enhanced algorithm for topological changes where the intersection of curves is

We consider a blow-up phenomenon for $ { \partial_t^2 u_ \varepsilon} $ $ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $ $ { = F(\partial_t u_ \varepsilon)}. $ The derivative of the solution $ \partial_t u_ \varepsilon $ blows-up on a curve $ t = T_ \varepsilon(x) $ if we impose some conditions on the initial values and the nonlinear term $ F $. We call $ T_ \varepsilon $ blow-up curve for $ { \partial_t^2

In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of object of interest and its representation by the signed-distance function (OF$ ^{dist} $ method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which

In this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal.The evolution is described by a system coupling a Nernst-Planck system for the ions concentrations with a Maxwell's equation of electrostatics governing the evolution of the electrostatic potential, a Navier-Stokes

Well-posedness classes for degenerate elliptic problems in $ {\mathbb R}^N $ under the form $ u = \Delta {{\varphi}}(x,u)+f(x) $, with locally (in $ u $) uniformly continuous nonlinearities, are explored. While we are particularly interested in the $ L^\infty $ setting, we also investigate about solutions in $ L^1_{loc} $ and in weighted $ L^1 $ spaces. We give some sufficient conditions in order that

In this survey, we are interested in the instability of flame fronts regarded as free interfaces. We successively consider a classical Arrhenius kinetics (thin flame) and a stepwise ignition-temperature kinetics (thick flame) with two free interfaces. A general method initially developed for thin flame problems subject to interface jump conditions is proving to be an effective strategy for smoother

The kinetic and potential energies for the damped wave equation $ \begin{equation} u''+2Bu'+A^2u = 0 \;\;\;\;\;\;({\rm DWE})\end{equation} $

The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

We consider a reaction-diffusion model of biological invasion in which the evolution of the population is governed by several parameters among them the intrinsic growth rate $ \mu(x) $. The knowledge of this growth rate is essential to predict the evolution of the population, but it is a priori unknown for exotic invasive species. We prove uniqueness and unconditional Lipschitz stability for the corresponding

We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities of type $ |u|^{p-1}u $, when the initial datas are in negative Sobolev spaces $ H_q^{-s}(\Omega) $, $ \Omega \subset \mathbb{R}^N $, $ s \in [0,2] $, $ q \in (1,\infty) $. Existence is for instance proved for $ q>\frac{N}{2}\left(\frac{1}{p-1}-\frac{s}{2}\right)^{-1} $.

In this article, we deal with the numerical immersed boundary-lattice Boltzmann method for simulation of the fluid-structure interaction problems in 2D. We consider the interaction of incompressible, Newtonian fluid in an isothermal system with an elastic fiber, which represents an immersed body boundary. First, a short introduction to the lattice Boltzmann and immersed boundary method is presented

We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between

A numerical method for solving diffusion problems on polyhedral meshes is presented. It is based on a finite volume approximation with the degrees of freedom located in the centers of computational cells. A numerical gradient is defined by a least-squares minimization for each cell, where we suggest a restricted form in the case of discontinuous diffusion coefficient. The flux balanced approximation

Here, we report a novel method of 3D image segmentation, using surface reconstruction from 3D point cloud data and 3D digital image information. For this task, we apply a mathematical model and numerical method based on the level set algorithm. This method solves surface reconstruction by the application of advection equation with a curvature term, which gives the evolution of an initial condition

In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains

We construct a data-driven dynamical system model for a macroscopic variable the Reynolds number of a high-dimensionally chaotic fluid flow by training its scalar time-series data. We use a machine-learning approach, the reservoir computing for the construction of the model, and do not use the knowledge of a physical process of fluid dynamics in its procedure. It is confirmed that an inferred time-series

We prove existence of $ L^1 $-weak solutions to the reaction-diffusion system obtained as a stationary version of the system arising for the evolution of concentrations in a reversible chemical reaction, coupled with space diffusion. This extends a previous result by the same authors where restrictive assumptions on the number of chemical species are removed.

We propose a time semi-discrete scheme for the Caginalp phase-field system with singular potentials and dynamic boundary conditions. The scheme is based on a time splitting which decouples the equations and on a convex splitting of the energy associated to the problem. The scheme is unconditionally uniquely solvable and the energy is nonincreasing if the time step is small enough. The discrete solution

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, $ N\geq 1 $, and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the

Autonomous sustained oscillations are ubiquitous in living and nonliving systems. As open systems, far from thermodynamic equilibrium, they defy entropic laws which mandate convergence to stationarity. We present structural conditions on network cycles which support global Hopf bifurcation, i.e. global bifurcation of non-stationary time-periodic solutions from stationary solutions. Specifically, we

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information

The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker–Planck equation can be

The paper presents a numerical study of the efficiency of the newly proposed far-field boundary simulations of wall-bounded, stably stratified flows. The comparison of numerical solutions obtained on large and truncated computational domain demonstrates how the solution is affected by the adopted far-field conditions. The mathematical model is based on Boussinesq approximation for stably stratified

Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization.First, we show that

A self-contained review is given for the development and current state of implicit constitutive modelling of viscoelastic response of materials in the context of strain-limiting theory.

Using rate-independent evolutions as a framework for elastoplasticity, an a posteriori bound for the error introduced by time stepping is established. A time adaptive algorithm is devised and tested in comparison to a method with constant time steps. Experiments show that a significant error reduction can be obtained using variable time steps.

We study a rate-independent system with non-convex energy in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized $ BV $-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact.

We prove that the spectrum of the linear delay differential equation $ x'(t) = A_{0}x(t)+A_{1}x(t-\tau_{1})+\ldots+A_{n}x(t-\tau_{n}) $ with multiple hierarchical large delays $ 1\ll\tau_{1}\ll\tau_{2}\ll\ldots\ll\tau_{n} $ splits into two distinct parts: the strong spectrum and the pseudo-continuous spectrum. As the delays tend to infinity, the strong spectrum converges to specific eigenvalues of

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorovich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of

We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates [44]. Computational experiments exploiting C1 finite elements are provided

We prove a global existence, uniqueness and regularity result for a two-species reaction-diffusion volume-surface system that includes nonlinear bulk diffusion and nonlinear (weak) cross diffusion on the active surface. A key feature is a proof of upper $ L^{\infty} $-bounds that exploits the entropic gradient structure of the system.

We prove existence of weak solutions of a Mullins-Sekerka equation on a surface that is coupled to diffusion equations in a bulk domain and on the boundary. This model arises as a sharp interface limit of a phase field model to describe the formation of liqid rafts on a cell membrane. The solutions are constructed with the aid of an implicit time discretization and tools from geometric measure theory

We consider a model for the motion of a phase interface in an elastic medium, for example, a twin boundary in martensite. The model is given by a semilinear parabolic equation with a fractional Laplacian as regularizing operator, stemming from the interaction of the front with its elastic environment. We show that the presence of randomly distributed, localized obstacles leads to a threshold phenomenon

We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($ \Gamma $-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $ \varepsilon $ and $ \tau $

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $ \Lambda $-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $ p $-Laplace operator with

Recent progress in the mathematical analysis of variational models for the plastic deformation of crystals in a geometrically nonlinear setting is discussed. The focus lies on the first time-step and on situations where only one slip system is active, in two spatial dimensions. The interplay of invariance under finite rotations and plastic deformation leads to the emergence of microstructures, which

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity (i.e. concentration gradient) term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are

The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of

We are devoted with fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications from partial differential equations are given to illustrate the abstract fractional degenerate differential problems.

The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem $ \left\{ \begin{array}{ll} - {\rm{div}} (a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ f(x,u) &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. $

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

In this paper, we define $ E_ \alpha(t^ \alpha A) $, where $ A $ is the generator of an uniformly bounded ($ C_0 $) semigroup and $ E_ \alpha(z) $ the Mittag-Leffler function. Since the mapping $ t\mapsto E_ \alpha(t^ \alpha A) $ has not the semigroup property, we cannot use the Trotter formula for representing the Feynman operator calculus. Thus for the Hamiltonian $ H_ \alpha = -\frac{{\hbar_ \alpha2}}{{2m}}\Delta

We study positive solutions to a steady state reaction diffusion equation arising in population dynamics, namely, $ \begin{equation*} \label{abs} \left\lbrace \begin{matrix}-\Delta u = \lambda u(1-u) ;\; x\in\Omega\\ \frac{\partial u}{\partial \eta}+\gamma\sqrt{\lambda}[(A-u)^2+\epsilon]u = 0; \; x\in\partial \Omega \end{matrix} \right. \end{equation*} $

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrödinger operators with sufficiently short-range interactions in general space dimensions.More specifically, assuming a sufficient power law decay of potentials, we derive the absence of

We study mixed hyperbolic systems with dynamic and Wentzell boundary conditions. The boundary condition contains a tangential operator which is strongly elliptic on the boundary. We prove results of generation of strongly continuous groups and well-posedness.