Mariya Ptashnyk | University of Dundee (original) (raw)
Papers by Mariya Ptashnyk
Multiscale Modeling & Simulation, 2015
In this paper we generalize the periodic unfolding method and the notion of two-scale convergence... more In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic (l-p) situation. The introduced methods allow us to consider a wider range of non-periodic microstructures. Using techniques of locally periodic homogenization we obtain macroscopic equations that are different from those derived by applying periodic homogenization techniques to problems posed in domains with changes in the microstructure represented by periodic coefficients depending on slow and fast variables. Using the method of locally periodic two-scale (l-t-s) convergence on oscillating surfaces and locally periodic (l-p) boundary unfolding operator we can analyze differential equations defined on boundaries of nonperiodic microstructures.
Mathematical Methods in the Applied Sciences, 2014
ABSTRACT With methods of potential theory, we develop a representation of a solution of the coupl... more ABSTRACT With methods of potential theory, we develop a representation of a solution of the coupled Stokes–Darcy model in a Lipschitz domain for boundary data in H−1/2. Copyright © 2014 John Wiley & Sons, Ltd.
In this paper, we focus on the Keller-Segel chemotaxis system in a random heterogeneous domain. W... more In this paper, we focus on the Keller-Segel chemotaxis system in a random heterogeneous domain. We assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two-scale convergence methods to derive the homogenized macroscopic equations. In establishing our results, we also derive a priori estimates for the Keller-Segel system that rely only on the boundedness of the coefficients; in particular, no differentiability assumption on the coefficients is required. Finally, we prove the convergence of a periodization procedure for approximating the homogenized asymptotic coefficients.
The aim of this thesis is to prove existence and uniqueness of weak solutions for some types of q... more The aim of this thesis is to prove existence and uniqueness of weak solutions for some types of quasilinear and nonlinear pseudoparabolic equations and for some types of quasilinear and nonlinear variational inequalities. The pseudoparabolic equations are characterized by the presence of mixed third order derivatives. Here the existence theory for degenerate parabolic equations is extended to the pseudoprabolic case, and degenerate pseudoparabolic equations with nonlinear integral operator are treated. Furthermore, quasilinear equations, posed on time intervals of the form (−∞, T], are considered. Some nonlinear pseudoparabolic equations are obtained as reduced form of systems of equations. To show existence, the Galerkin and Rothe methods are used. The system of the degenerate equations, where the term ∂ t u is replace by ∂ t b(u), is solved using the monotonicity and gradient assumptions on the nonlinear function b. The discretization along characteristics is applied to equations with convection. The existence of solutions of variational inequalities is proved by a penalty method; here an inequality is replaced by an equation with an added penalty operator. The uniqueness follows from the monotonicity of the differential operators. In the case of nonlinear pseudoparabolic equations, the uniqueness can be shown for regular solutions only. The needed regularity is shown for two dimensional domains. Zusammenfassung Thema dieser Arbeit sind sowohl quasilineare und nichtlineare pseudoparabolische Glei chungen als auch solche Variationsungleichungen. Pseudoparabolische Gleichungen sind durch Auftreten von gemischten Ableitungen von dritter Ordnung charakterisiert. Für einige Typen solcher Gleichungen bzw. Ungleichungen wird in dieser Arbeit die Lösbarkeit gezeigt. In fast allen Fällen kann auch die Eindeutigkeit bewiesen werden. Die Existenztheorie für entartete parabolische Gleichungen wird auf den Fall pseudoparabolischer Gleichungen erweitert. Entartete Gleichungen mit nichtlinearen Integraloperatoren werden ebenfalls behandelt. Außerdem werden quasilineare Gleichungen für Zeitintervalle der Form (−∞, T] betrachtet. Einige nichtlineare pseudoparabolische Gleichungen erhält man durch Reduktion von Systemen. Für den Beweis der Existenz werden die Rothe-und Galerkin-Methoden benutzt. Die Existenz von Lösungen des Systems entarteter Gleichungen ist unter Annahme der Monotonie und der Rotationsfreiheit der nichtlineare Funktion gezeigt; genauer, die nichtlineare Funktion ist ein Gradient. Die Gleichungen mit Konvektion werden hier entlang der Charakteristiken diskretisiert. Die Existenz von Lösungen für Variationsungleichungen ist mit Hilfe der Strafterm-Methode gezeigt. Die Eindeutigkeit der Lösung folgt aus der Monotonie der Operatoren. Die Eindeutigkeit der Lösung der nichtlinearen Gleichungen ist nur für reguläre Lösungen bewiesen, wobei schwache Lösungen in zwei Dimension schon diese Regularität besitzen.
Journal of Mathematical Biology, 2014
Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system ... more Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by Send offprint requests to: A. Chavarría-Krauser This work was supported by: German Research Foundation grant number CH 958/1-1, Excellence Initiative II Heidelberg University "Mobilitätsmaßnahmen im Rahmen internationaler Forschungskooperationen 2013-2014" project number D.80100/13.009, and EPSRC grant number EP/K036521/1.
Ukrainian Mathematical Journal, 1999
We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in va... more We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variable t. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior as t --~ -,,o.
Mathematical Modelling of Natural Phenomena, 2013
Water flow in plant tissues takes place in two different physical domains separated by semipermea... more Water flow in plant tissues takes place in two different physical domains separated by semipermeable membranes: cell insides and cell walls. The assembly of all cell insides and cell walls are termed symplast and apoplast, respectively. Water transport is pressure driven in both, where osmosis plays an essential role in membrane crossing. In this paper, a microscopic model of water flow and transport of an osmotically active solute in a plant tissue is considered. The model is posed on the scale of a single cell and the tissue is assumed to be composed of periodically distributed cells. The flow in the symplast can be regarded as a viscous Stokes flow, while Darcy's law applies in the porous apoplast. Transmission conditions at the interface (semipermeable membrane) are obtained by balancing the mass fluxes through the interface and by describing the protein mediated transport as a surface reaction. Applying homogenization techniques, macroscopic equations for water and solute transport in a plant tissue are derived. The macroscopic problem is given by a Darcy law with a force term proportional to the difference in concentrations of the osmotically active solute in the symplast and apoplast; i.e. the flow is also driven by the local concentration difference and its direction can be different than the one prescribed by the pressure gradient.
Ukrainian Mathematical Journal, 1998
We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a var... more We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable t. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as t --~ ~,.
Multiscale Modeling & Simulation, 2013
The introduced notion of locally-periodic two-scale convergence allows to average a wider range o... more The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in H 1 (Ω) are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.
SIAM Journal on Mathematical Analysis, 2008
We study the problem of diffusive transport of biomolecules in the intercellular space, modeled a... more We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of twoscale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors of biological cells and compare the derived model with those previously considered.
SIAM Journal on Applied Mathematics, 2010
In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into ... more In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of the plant-soil interaction when trying to understand solute losses from the soil.
Nonlinear Analysis: Real World Applications, 2010
A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak a... more A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak acid which dissociates into ions in the aqueous cell compartments. A microscopic model is defined by diffusion-reaction equations and a Poisson equation for a given charge distribution. The microscopic properties of the plant cell were taken into account through oscillating coefficients in the
Nonlinear Analysis: Real World Applications, 2010
ABSTRACT In this article the process of nutrient uptake by a single root branch is studied. We co... more ABSTRACT In this article the process of nutrient uptake by a single root branch is studied. We consider diffusion and active transport of nutrients dissolved in water. The uptake happens on the surface of thin root hairs distributed periodically and orthogonal to the root surface. Water velocity is defined by the Stokes equations. We derive a macroscopic model for nutrient uptake by a hairy-root from microscopic descriptions using homogenization techniques. The macroscopic model consists of a reaction-diffusion equation in the domain with hairs and a diffusion-convection equation in the domain without hairs. The macroscopic water velocity is described by the Stokes system in the domain without hairs, with no-slip condition on the boundary between domain with hairs and free fluid.
Nonlinear Analysis: Theory, Methods & Applications, 2007
The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂ ... more The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂ t u is replace by ∂ t b(u), with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitzcontinuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous.
New Phytologist, 2010
Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but ... more Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes.
Journal of Theoretical Biology, 2012
Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed o... more Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed on nonplanar surfaces is investigated. First, we present a continuous model of lateral membrane dynamics, described by a nonlinear PDE of fourth order. We then prove the existence and uniqueness of solutions of the presented model and simulate membrane dynamics using a finite element approach. Adopting a truly multiscale approach, we use dissipative particle dynamics (DPD) to parameterize the continuous model, i.e. to derive a corresponding macroscopic model.
IMA Journal of Applied Mathematics, 2007
The existence of solutions of pseudoparabolic equations with convection by using discretization a... more The existence of solutions of pseudoparabolic equations with convection by using discretization along characteristics is shown. The uniqueness of the solution of a pseudoparabolic equation is proved for a linear elliptic part and for a space dimension N [<=] 4.
ESAIM: Control, Optimisation and Calculus of Variations, 2012
Ukrainian Mathematical Journal, 2002
We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existe... more We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existence and uniqueness of a solution of this inequality with a zero initial condition.
Multiscale Modeling & Simulation, 2015
In this paper we generalize the periodic unfolding method and the notion of two-scale convergence... more In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic (l-p) situation. The introduced methods allow us to consider a wider range of non-periodic microstructures. Using techniques of locally periodic homogenization we obtain macroscopic equations that are different from those derived by applying periodic homogenization techniques to problems posed in domains with changes in the microstructure represented by periodic coefficients depending on slow and fast variables. Using the method of locally periodic two-scale (l-t-s) convergence on oscillating surfaces and locally periodic (l-p) boundary unfolding operator we can analyze differential equations defined on boundaries of nonperiodic microstructures.
Mathematical Methods in the Applied Sciences, 2014
ABSTRACT With methods of potential theory, we develop a representation of a solution of the coupl... more ABSTRACT With methods of potential theory, we develop a representation of a solution of the coupled Stokes–Darcy model in a Lipschitz domain for boundary data in H−1/2. Copyright © 2014 John Wiley & Sons, Ltd.
In this paper, we focus on the Keller-Segel chemotaxis system in a random heterogeneous domain. W... more In this paper, we focus on the Keller-Segel chemotaxis system in a random heterogeneous domain. We assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two-scale convergence methods to derive the homogenized macroscopic equations. In establishing our results, we also derive a priori estimates for the Keller-Segel system that rely only on the boundedness of the coefficients; in particular, no differentiability assumption on the coefficients is required. Finally, we prove the convergence of a periodization procedure for approximating the homogenized asymptotic coefficients.
The aim of this thesis is to prove existence and uniqueness of weak solutions for some types of q... more The aim of this thesis is to prove existence and uniqueness of weak solutions for some types of quasilinear and nonlinear pseudoparabolic equations and for some types of quasilinear and nonlinear variational inequalities. The pseudoparabolic equations are characterized by the presence of mixed third order derivatives. Here the existence theory for degenerate parabolic equations is extended to the pseudoprabolic case, and degenerate pseudoparabolic equations with nonlinear integral operator are treated. Furthermore, quasilinear equations, posed on time intervals of the form (−∞, T], are considered. Some nonlinear pseudoparabolic equations are obtained as reduced form of systems of equations. To show existence, the Galerkin and Rothe methods are used. The system of the degenerate equations, where the term ∂ t u is replace by ∂ t b(u), is solved using the monotonicity and gradient assumptions on the nonlinear function b. The discretization along characteristics is applied to equations with convection. The existence of solutions of variational inequalities is proved by a penalty method; here an inequality is replaced by an equation with an added penalty operator. The uniqueness follows from the monotonicity of the differential operators. In the case of nonlinear pseudoparabolic equations, the uniqueness can be shown for regular solutions only. The needed regularity is shown for two dimensional domains. Zusammenfassung Thema dieser Arbeit sind sowohl quasilineare und nichtlineare pseudoparabolische Glei chungen als auch solche Variationsungleichungen. Pseudoparabolische Gleichungen sind durch Auftreten von gemischten Ableitungen von dritter Ordnung charakterisiert. Für einige Typen solcher Gleichungen bzw. Ungleichungen wird in dieser Arbeit die Lösbarkeit gezeigt. In fast allen Fällen kann auch die Eindeutigkeit bewiesen werden. Die Existenztheorie für entartete parabolische Gleichungen wird auf den Fall pseudoparabolischer Gleichungen erweitert. Entartete Gleichungen mit nichtlinearen Integraloperatoren werden ebenfalls behandelt. Außerdem werden quasilineare Gleichungen für Zeitintervalle der Form (−∞, T] betrachtet. Einige nichtlineare pseudoparabolische Gleichungen erhält man durch Reduktion von Systemen. Für den Beweis der Existenz werden die Rothe-und Galerkin-Methoden benutzt. Die Existenz von Lösungen des Systems entarteter Gleichungen ist unter Annahme der Monotonie und der Rotationsfreiheit der nichtlineare Funktion gezeigt; genauer, die nichtlineare Funktion ist ein Gradient. Die Gleichungen mit Konvektion werden hier entlang der Charakteristiken diskretisiert. Die Existenz von Lösungen für Variationsungleichungen ist mit Hilfe der Strafterm-Methode gezeigt. Die Eindeutigkeit der Lösung folgt aus der Monotonie der Operatoren. Die Eindeutigkeit der Lösung der nichtlinearen Gleichungen ist nur für reguläre Lösungen bewiesen, wobei schwache Lösungen in zwei Dimension schon diese Regularität besitzen.
Journal of Mathematical Biology, 2014
Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system ... more Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been modeled by a system of ordinary differential equations based on the uptake of zinc, expression of a transporter protein and the interaction between an activator and inhibitor. For certain parameter choices the steady state of this model becomes unstable upon variation in the external zinc concentration. Numerical results show periodic orbits emerging between two critical values of the external zinc concentration. Here we show the existence of a global Hopf bifurcation with a continuous family of stable periodic orbits between two Hopf bifurcation points. The stability of the orbits in a neighborhood of the bifurcation points is analyzed by deriving the normal form, while the stability of the orbits in the global continuation is shown by calculation of the Floquet multipliers. From a biological point of view, stable periodic orbits lead to potentially toxic zinc peaks in plant cells. Buffering is believed to be an efficient way to deal with strong transient variations in zinc supply. We extend the model by Send offprint requests to: A. Chavarría-Krauser This work was supported by: German Research Foundation grant number CH 958/1-1, Excellence Initiative II Heidelberg University "Mobilitätsmaßnahmen im Rahmen internationaler Forschungskooperationen 2013-2014" project number D.80100/13.009, and EPSRC grant number EP/K036521/1.
Ukrainian Mathematical Journal, 1999
We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in va... more We consider a nonlinear pseudoparabolic variational inequality in a tube domain semibounded in variable t. Under certain conditions imposed on coefficients of the inequality, we prove the theorems of existence and uniqueness of a solution without any restriction on its behavior as t --~ -,,o.
Mathematical Modelling of Natural Phenomena, 2013
Water flow in plant tissues takes place in two different physical domains separated by semipermea... more Water flow in plant tissues takes place in two different physical domains separated by semipermeable membranes: cell insides and cell walls. The assembly of all cell insides and cell walls are termed symplast and apoplast, respectively. Water transport is pressure driven in both, where osmosis plays an essential role in membrane crossing. In this paper, a microscopic model of water flow and transport of an osmotically active solute in a plant tissue is considered. The model is posed on the scale of a single cell and the tissue is assumed to be composed of periodically distributed cells. The flow in the symplast can be regarded as a viscous Stokes flow, while Darcy's law applies in the porous apoplast. Transmission conditions at the interface (semipermeable membrane) are obtained by balancing the mass fluxes through the interface and by describing the protein mediated transport as a surface reaction. Applying homogenization techniques, macroscopic equations for water and solute transport in a plant tissue are derived. The macroscopic problem is given by a Darcy law with a force term proportional to the difference in concentrations of the osmotically active solute in the symplast and apoplast; i.e. the flow is also driven by the local concentration difference and its direction can be different than the one prescribed by the pressure gradient.
Ukrainian Mathematical Journal, 1998
We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a var... more We consider a pseudoparabolic variational inequality in a cylindrical domain semibounded in a variable t. Under certain conditions imposed on the coefficients of the inequality, we prove theorems on the unique existence of a solution for a class of functions with exponential growth as t --~ ~,.
Multiscale Modeling & Simulation, 2013
The introduced notion of locally-periodic two-scale convergence allows to average a wider range o... more The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in H 1 (Ω) are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.
SIAM Journal on Mathematical Analysis, 2008
We study the problem of diffusive transport of biomolecules in the intercellular space, modeled a... more We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of twoscale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors of biological cells and compare the derived model with those previously considered.
SIAM Journal on Applied Mathematics, 2010
In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into ... more In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of the plant-soil interaction when trying to understand solute losses from the soil.
Nonlinear Analysis: Real World Applications, 2010
A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak a... more A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak acid which dissociates into ions in the aqueous cell compartments. A microscopic model is defined by diffusion-reaction equations and a Poisson equation for a given charge distribution. The microscopic properties of the plant cell were taken into account through oscillating coefficients in the
Nonlinear Analysis: Real World Applications, 2010
ABSTRACT In this article the process of nutrient uptake by a single root branch is studied. We co... more ABSTRACT In this article the process of nutrient uptake by a single root branch is studied. We consider diffusion and active transport of nutrients dissolved in water. The uptake happens on the surface of thin root hairs distributed periodically and orthogonal to the root surface. Water velocity is defined by the Stokes equations. We derive a macroscopic model for nutrient uptake by a hairy-root from microscopic descriptions using homogenization techniques. The macroscopic model consists of a reaction-diffusion equation in the domain with hairs and a diffusion-convection equation in the domain without hairs. The macroscopic water velocity is described by the Stokes system in the domain without hairs, with no-slip condition on the boundary between domain with hairs and free fluid.
Nonlinear Analysis: Theory, Methods & Applications, 2007
The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂ ... more The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term ∂ t u is replace by ∂ t b(u), with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitzcontinuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b to be Lipschitz continuous.
New Phytologist, 2010
Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but ... more Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes.
Journal of Theoretical Biology, 2012
Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed o... more Combining different theoretical approaches, curvature modulated sorting in lipid bilayers fixed on nonplanar surfaces is investigated. First, we present a continuous model of lateral membrane dynamics, described by a nonlinear PDE of fourth order. We then prove the existence and uniqueness of solutions of the presented model and simulate membrane dynamics using a finite element approach. Adopting a truly multiscale approach, we use dissipative particle dynamics (DPD) to parameterize the continuous model, i.e. to derive a corresponding macroscopic model.
IMA Journal of Applied Mathematics, 2007
The existence of solutions of pseudoparabolic equations with convection by using discretization a... more The existence of solutions of pseudoparabolic equations with convection by using discretization along characteristics is shown. The uniqueness of the solution of a pseudoparabolic equation is proved for a linear elliptic part and for a space dimension N [<=] 4.
ESAIM: Control, Optimisation and Calculus of Variations, 2012
Ukrainian Mathematical Journal, 2002
We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existe... more We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existence and uniqueness of a solution of this inequality with a zero initial condition.