# Piotr KalitaJagiellonian University | UJ · Faculty of Mathematics and Computer Science

Piotr Kalita

associate professor

## About

88

Publications

7,837

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661

Citations

Citations since 2016

Introduction

Piotr Kalita currently works at the Faculty of Mathematics and Computer Science, Jagiellonian University.

Additional affiliations

October 2005 - present

## Publications

Publications (88)

If the semigroup is slowly non-dissipative, i.e., its {\color{black} solutions} can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors - the counterpart of the classical notion of global attractors. We continue the development of this theory started by Chepyzhov and Goritskii...

In this paper we study in detail the structure of the global attractor for a generalized Lotka-Volterra system with Volterra--Lyapunov stable structural matrix. We provide the full characterization of this structure and we show that it coincides with the invasion graph as recently introduced in [15]. We also study the stability of the structure wit...

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah–Struwe solutions, which satisfy the Strichartz estimates and coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, unifor...

We study the elliptic inclusion given in the following divergence form $$\begin{aligned}&-\mathrm {div}\,A(x,\nabla u) \ni f\quad \mathrm {in}\quad \Omega ,\\&u=0\quad \mathrm {on}\quad \partial \Omega . \end{aligned}$$As we assume that \(f\in L^1(\Omega )\), the solutions to the above problem are understood in the renormalized sense. We also assum...

We study the non-autonomous weakly damped wave equation with subquintic growth condition on the nonlinearity. Our main focus is the class of Shatah--Struwe solutions, which satisfy the Strichartz estimates and are coincide with the class of solutions obtained by the Galerkin method. For this class we show the existence and smoothness of pullback, u...

We investigate the upper bound on the vertical heat transport in the fully 3D Rayleigh–Bénard convection problem at the infinite Prandtl number for a micropolar fluid. We obtain a bound, given by the cube root of the Rayleigh number, with a logarithmic correction. The derived bound is compared with the optimal known one for the Newtonian fluid. It...

In this paper we present some mutual relations between semigroup theory in the context of the theory of infinite dimensional dynamical systems and the mathematical theory of hydrodynamics. These mutual relations prove to be very fruitful, enrich both fields and help to understand behaviour of solutions of both infinite dimensional dynamical systems...

We propose a method to integrate dissipative PDEs rigorously forward in time with the use of Finite Element Method (FEM). The technique is based on the Galerkin projection on the FEM space and estimates on the residual terms. The technique is illustrated on a periodically forced one-dimensional Burgers equation with Dirichlet conditions. For two pa...

We study the elliptic inclusion given in the following divergence form \begin{align*} & -\mathrm{div}\, A(x,\nabla u) \ni f\quad \mathrm{in}\quad \Omega, & u=0\quad \mathrm{on}\quad \partial \Omega. \end{align*} As we assume that $f\in L^1(\Omega)$, the solutions to the above problem are understood in the renormalized sense. We also assume nonstand...

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios...

The verification of efficiency of the diffusive-type model of neurotransmitter transport dynamics inside the presynaptic bouton, including the spatial aspect of the transport, is the main objective of the simulations described in this paper. Finite element and finite difference methods were applied to solve the model numerically. The bouton was rep...

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$
on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstra...

Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attra...

We consider the Rayleigh--B\'{e}nard problem for the three--dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor $\mathcal{A}^K$ corresponding to weak solutions, for every micropolar parameter $K\geq 0$ denoting the deviation of...

This paper is devoted to the study of global attractors for problems with state-dependent impulses and possible nonuniqueness of solutions. We provide the criteria under which there exists the global attractor, being on one hand an invariant set, and on the other hand given by the difference of the minimal compact attracting set and the impulsive s...

We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our es...

The two-dimensional Rayleigh–Bénard problem for a thermomicropolar fluids model is considered. The existence of suitable weak solutions which may not be unique, and the existence of the unique strong solution are proved. The global attractor for the m-semiflow associated with weak solutions and the global attractor for semiflow associated with stro...

We study the non-autonomously forced Burgers equation $$ u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t) $$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \R$. Moreover...

We present an approach based on the Kakutani–Ky Fan–Glicksberg fixed point theorem and a truncation argument, to prove the existence of strong solutions for two boundary value problems for the Duffing-type ordinary differential equation with multivalued terms. As a special case we recover the original Duffing equation.

In this work we prove the lower and upper semicontinuity of pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation on a bounded domain Ω⊂R3ϵutt+ut−Δu=fϵ(t,u).
For each ϵ>0, this equation has uniform, pullback, and cocycle attractors in H01(Ω)×L2(Ω) and for ϵ=0 the limit parabolic equationut−Δu...

Using the time approximation method we obtain the existence of a weak solution for the dynamic contact problem with damping and a non-convex stored elastic energy function. On the contact boundary we assume the normal compliance law and the generalization of the Coulomb friction law which allows for non-monotone dependence of the friction force on...

We study the stationary incompressible flow of a generalized Newtonian fluid described by a nonlinear multivalued maximal monotone constitutive law and a multivalued nonmonotone frictional boundary condition. We provide results on the existence and uniqueness of a solution to the variational form of the problem. When the multivalued laws are of a s...

The present paper represents a continuation of [3]. There, we studied a new class of variational inequalities involving a pseudomonotone univalued operator and a multivalued operator, for which we obtained an existence result, among others. In the current paper we prove that this result remains valid under significantly weaker assumption on the mul...

The Navier-Stokes equations, which describe the movement of fluids, are an important source of topics for scientific research, technological development and innovation. Fluids are part of the systems of nature, life and society. Involved in many processes and phenomena of everyday life as an active part of the interactions within the hosting system...

The Navier-Stokes equations, which describe the movement of fluids, are an important source of topics for scientific research, technological development and innovation. Fluids are part of the systems of nature, life and society. Involved in many processes and phenomena of everyday life as an active part of the interactions within the hosting system...

This paper refers strongly to mathematical modeling of diffusive process in a presynaptic bouton. Creation of a robust three-dimensional model of the bouton geometry is the topic of the paper. Such a model is necessary for partial differential equations that describe the aforementioned flows. The proposed geometric model is based on ultrathin secti...

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well a...

A generalization of the Oberbeck–Boussinesq model consisting of a system of steady state multivalued partial differential equations for incompressible, generalized Newtonian of the p-power type, viscous flow coupled with the nonlinear heat equation is studied in a bounded domain. The existence of a weak solution is proved by combining the surjectiv...

In this article we study the operator version of a first order in time partial differential inclusion as well as its time discretization obtained by an implicit Euler scheme. This technique, known as the Rothe method, yields the semidiscrete trajectories that are proved to converge to the solution of the original problem. While both the time contin...

In this chapter we give an overview of the equations of classical hydrodynamics. We provide their derivation, comment on the stress tensor, and thermodynamics, finally we present some elementary properties and also some exact solutions of the Navier–Stokes equations.

In this chapter we introduce some basic notions from the theory of the Navier–Stokes equations: the function spaces H, V, and V ′, the Stokes operator A with its domain D(A) in H, and the bilinear form B. We apply the Galerkin method and fixed point theorems to prove the existence of solutions of the nonlinear stationary problem, and we consider pr...

We start this chapter from necessary background on the theory of fractal dimension. Next, we formulate and study a problem which models the two-dimensional boundary driven shear flow in lubrication theory. After the derivation of the energy dissipation rate estimate and a version of Lieb–Thirring inequality we provide an estimate from above on the...

In this chapter we consider two-dimensional nonstationary incompressible Navier–Stokes shear flows with nonmonotone and multivalued leak boundary conditions on a part of the boundary of the flow domain. Our considerations are motivated by feedback control problems for fluid flows in domains with semipermeable walls and membranes and by the theory o...

This chapter is devoted to the study of three-dimensional nonstationary Navier–Stokes equations with the multivalued frictional boundary condition. We use the formalism of evolutionary systems to prove the existence of weak global attractor for the studied problem.

In this chapter we consider the problem of existence and finite dimensionality of the pullback attractor for a class of two-dimensional turbulent boundary driven flows which naturally appear in lubrication theory. We generalize here the results from Chap. 9 to the non-autonomous problem.

This chapter provides, for the convenience of the reader, an overview of the whole book, first of its structure and then of the content of the individual chapters.

In this chapter we consider the three-dimensional stationary Navier–Stokes equations with multivalued friction law boundary conditions on a part of the domain boundary. We formulate two existence theorems for the formulated problem. The first one uses the Kakutani–Fan–Glicksberg fixed point theorem, and the second one, with the relaxed assumptions,...

In this chapter we introduce the basic preliminary mathematical tools to study the Navier–Stokes equations, including results from linear and nonlinear functional analysis as well as the theory of function spaces. We present, in particular, some of the most frequently used in the sequel embedding theorems and differential inequalities.

In this chapter we prove the existence of invariant measures associated with two-dimensional autonomous Navier–Stokes equations. Then we introduce the notion of a stationary statistical solution and prove that every invariant measure is also such a solution.

In this chapter we consider two examples of contact problems. First, we study the problem of time asymptotics for a class of two-dimensional turbulent boundary driven flows subject to the Tresca friction law which naturally appears in lubrication theory. Then we analyze the problem with the generalized Tresca law, where the friction coefficient can...

This chapter is devoted to constructions of invariant measures and statistical solutions for non-autonomous Navier–Stokes equations in bounded and certain unbounded domains in \(\mathbb{R}^{2}\).After introducing some basic notions and results concerning attractors in the context of the Navier–Stokes equations, we construct the family of probabilit...

In this chapter we consider further non-autonomous and multivalued evolution problems, this time in the frame of the theory of pullback attractors for multivalued processes.

In this chapter we study a typical problem from the theory of lubrication, namely, the Stokes flow in a thin three-dimensional domain \(\varOmega ^{\varepsilon }\), \(\varepsilon> 0\). We assume the Fourier boundary condition (only the friction part) at the top surface and a nonlinear Tresca interface condition at the bottom one.

In this chapter we study the time asymptotics of solutions to the two-dimensional Navier–Stokes equations. In the first two sections we prove two properties of the equations in a bounded domain, concerning the existence of determining modes and nodes. Then we study the equations in an unbounded domain, in the framework of the theory of infinite dim...

This chapter contains some basic facts about solutions of nonstationary Navier–Stokes equations

In this paper we study a new multidimensional mixed-kinetic adsorption model which consists of a nonlinear evolution system of two parabolic partial differential equations: a convective diffusion equation for the bulk surfactant concentration in a bounded domain and a surface diffusion equation for its surface concentration on a compact Lipschitz m...

This volume is devoted to the study of the Navier–Stokes equations, providing a comprehensive reference for a range of applications: from students to engineers and mathematicians involved in research on fluid mechanics, dynamical systems, and mathematical modeling. Equipped with only a basic knowledge of calculus, functional analysis, and partial d...

We consider the evolution of weak vanishing viscosity solutions to the
critically dissipative surface quasi-geostrophic equation. Due to the possible
non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical
system and prove the existence of a global attractor of optimal Sobolev
regularity. To achieve this, we derive a new Sobo...

In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational form...

We consider a class of stationary subdifferential inclusions in a reflexive Banach space. We reformulate the problem in terms of a variational inequality with multivalued term and prove an existence result using the Kakutani-Fan-Glicksberg fixed point theorem. This approach allows to consider, in a natural way, a dual variational formulation of the...

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb's law of dry friction. In order to approximate the contact conditions, we consider a re...

We study the stationary two-dimensional incompressible flow of non-Newtonian fluid governed by a nonlinear constitutive law and with a multivalued nonmonotone subdifferential frictional boundary condition. We provide an abstract result on existence of solution to an operator inclusion modeling the flow phenomenon. We prove a theorem on existence an...

A method is proposed to deal with some multivalued processes with weak continuity properties. An application to a nonautonomous contact problem for the Navier–Stokes flow with nonmonotone multivalued frictional boundary condition is presented.

We prove an existence and uniqueness result for a class of subdifferential inclusions which involve a history-dependent operator. Then we specialize this result in the study of a class of history-dependent hemivariational inequalities. Problems of such kind arise in a large number of mathematical models which describe quasistatic processes of conta...

We consider two classes of evolution contact problems on two dimensional domains governed by first and second order evolution equations, respectively. The contact is represented by multivalued and nonmonotone boundary conditions that are expressed by means of Clarke subdifferentials of certain locally Lipschitz and semiconvex potentials. For both p...

We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with normal compliance and unilateral constraint, associated to a rate slip-dependent version of Coulomb’s law of dry friction. In order to approximate the contact conditions, we consider a re...

The paper deals with the convergence analysis of the semidiscrete Rothe scheme for the parabolic variational–hemivariational inequality with the nonlinear pseudomonotone elliptic operator. The problem involves both a discontinuous and nonmonotone multivalued term as well as a monotone term with potentials which assume infinite values and hence are...

We discuss the existence of pullback attractors for multivalued dynamical
systems on metric spaces. Such attractors are shown to exist without any
assumptions in terms of continuity of the solution maps, based only on
minimality properties with respect to the notion of pullback attraction. When
invariance is required, a very weak closed graph condi...

In this paper the sensitivity of optimal solutions to control problems described by second order evolution subdifferential inclusions under perturbations of state relations and of cost functionals is investigated. First we establish a new existence result for a class of such inclusions. Then, based on the theory of sequential \(\Gamma \) -convergen...

In this paper we show the convergence of a semidiscrete time stepping
\theta-scheme on a time grid of variable length to the solution of parabolic
operator di?erential inclusion in the framework of evolution triple. The
multifunction is assumed to be strong-weak upper-semicontinuous and to have
nonempty, closed and convex values, while the quasilin...

A method is proposed to deal with some multivalued semiflows with weak
continuity properties. An application to the reaction-diffusion problems with
nonmonotone multivalued semilinear boundary condition and nonmonotone
multivalued semilinear source term is presented.

We consider two-dimensional nonstationary Navier-Stokes shear flow with
multivalued and nonmonotone boundary conditions on a part of the boundary of
the flow domain. We prove the existence of global in time solutions of the
considered problem which is governed by a partial differential inclusion with a
multivalued term in the form of Clarke subdiff...

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone...

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin metho...

The paper is devoted to the Galerkin method and Finite Element Method for a stationary variational-hemivariational inequality modelling unilateral adhesive and frictionless contact of an elastic body with a foundation. Adhesion is modelled by a simplified Winkler-type law. An abstract theorem on the convergence of the Galerkin method for a class of...

The modulation of a signal that is transmitted in the nerve system takes place in chemical synapses. This article focuses on the phenomena undergone in the presynaptic part of the synapse. A diffusion–reaction type model based on the partial differential equation is proposed. Through an averaging procedure this model is reduced to a model based on...

We study optimal control of problem governed by a coupled system of hemivariational inequality (HVI) and ordinary differential equation (ODE). The problem models viscoelastic, adhesive contact between a body and a foundation. We employ the Kelvin-Voigt viscoelastic law and consider the general nonmonotone and multivalued subdifferential boundary co...

This paper deals with regularity of solutions to the abstract operator version of parabolic partial differential inclusion of the form u′(t)+Au(t)+ι⁎∂J(ιu(t))∋f(t)u′(t)+Au(t)+ι⁎∂J(ιu(t))∋f(t) with the multivalued term given in the form of Clarke subdifferential of a locally Lipschitz functional J. Using the Rothe method, it is shown that under appr...

This article presents the convergence analysis of a sequence of piecewise
constant and piecewise linear functions obtained by the Rothe method to the
solution of the first order evolution partial differential inclusion
$u'(t)+Au(t)+\iota^*\partial J(\iota u(t))\ni f(t)$, where the multivalued term
is given by the Clarke subdifferential of a locally...

In this article we consider the asymptotic behavior of solutions to second-order evolution inclusions with the boundary multivalued term and , where A is a (possibly) nonlinear coercive and pseudomonotone operator, B is linear, continuous, symmetric and coercive, is the trace operator and J is a locally Lipschitz integral functional with ∂ denoting...

Neurotransmitters in the terminal bouton of a presynaptic neuron are stored in vesicles, which diffuse in the cytoplasm and, after a stimulation signal is received, fuse with the membrane and release its contents into the synaptic cleft. It is commonly assumed that vesicles belong to three pools whose content is gradually exploited during the stimu...

In this paper a mathematical description of a presynaptic episode of slow synaptic neuropeptide transport is proposed. Two interrelated mathematical models, one based on a system of reaction diffusion partial differential equations and another one, a compartment type, based on a system of ordinary differential equations (ODE) are formulated. Proces...

In this paper a methodology of mathematical description of the synthesis, storage and release of the neurotransmitter during the fast synaptic transport is presented. The proposed model is based on the initial and boundary value problem for a parabolic nonlinear partial differential equation (PDE). Presented approach enables to express space and ti...

The article presents the up to date review and discussion of approaches used to express mechanical behavior of artery walls.
The physiology of artery walls and its relation to the models is discussed. Presented models include the simplest 0d and 1d
ones but emphasis is put to the most sophisticated approaches which are based on the theory of 3d non...

We present an algorithm for intelligent prediction of user requests in a system based on the services hosted by independent
providers. Data extracted from requests is organized in a dynamically changing graph representing dependencies between operations
and input arguments as well as between groups of arguments mutually coexisting in requests. The...

The paper focuses on models that express the phenomenon of pulse wave propagation in the human arterial system. Physical nonlinearities in both wall and blood tissues were included. Models are theoretically verified and the simulation results are compared with medical measurement data.

The paper presents mathematical and numerical models of the blood flow in human arteries. We describe selected modelling tech-niques for the mechanical phenomena occurring in the arteries: blood flow, displacement of the wall and the fluid-structure interaction be-tween the blood and the wall. The paper concentrates on the theoreti-cal results show...

In this paper we present different shell models and discuss their applicability to model the mechanical behavior of arterial walls. Presented models are: stationary and evolutionary, linear and nonlinear, membrane dominated and bending dominated. Special focus is put on the physically nonlinear stationary models which were first presented by the au...

In this paper we use the theory of monotone operators to generalize the linear shell model presented in (Blouza and Le Dret, 1999) to a class of physically nonlinear models. We present a family of nonlinear constitutive equations, for which we prove the existence and uniqueness of the solution of the presented nonlinear model, as well as the conver...

## Projects

Projects (2)

The aim of the work is to analyze the process of neurotransmitter (NT) synthesis, diffusion and exocytosis. The first stage comprises the synthesis, diffusion and release of NT in a presynaptic bouton of an axon nerve terminal.