System Of Evolution Equations Research Articles

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of implicit constitutive relations we establish the existence and uniqueness of a global-in-time large-data weak solution. We then focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises, which is that the Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.

Plane longitudinal waves in a fluid-saturated porous medium with a nonlinear relationship between deformations and displacements of the liquid phase

A mathematical model is presented to describe the propagation of a plane longitudinal wave in a fluid saturated porous medium, taking into account the geometric nonlinearity of the liquid component of the medium. The nonlinear relationship between deformations and displacements refines the classical Biot’s theory, within the framework of which a fluid saturated porous medium is considered. Evolutionary equations for the displacements of the skeleton of the medium and fluid in the pores are obtained. It is shown that, if the liquid is confined in the pores, then the wave propagation is described by an equation that generalizes the well-known Burgers equation and has a solution in the form of a stationary shock wave resulting from mutual compensation of the effects of nonlinearity and dissipation. The dependence of the width of the shock wave front on the viscosity of the fluid saturating the pores and the shock wave amplitude is determined. As the viscosity coefficient increases, the wave profile becomes steeper, i.e., the wave front width decreases. With an increase in the wave amplitude, the front width can either increase or decrease, depending on the other parameters of the original system. The limiting cases of the obtained generalized Burgers equation are analyzed with respect to the fluid viscosity parameter. If the liquid flows freely in the pores, then the system of evolutionary equations is reduced to a single equation of a simple wave, i.e., the propagation of a plane longitudinal wave in a porous medium can be described by the well-known equation of nonlinear wave dynamics - the Riemann equation. The equation describes the nonlinear waves, which are characterized by a steepening of the leading edge with subsequent overturning resulting from the growth of nonlinear effects in the absence of compensating factors such as dispersion and dissipation.

Some Approximate Solutions of the Dynamic Problem of Axisymmetric Shock Deformation of a Previously Unstressed Incompressible Elastic Medium

Approximate theoretical solutions are presented for the boundary value problem of the shock load on the boundary of a circular cylindrical cavity in the space occupied by an incompressible elastic medium undeformed previously. We assume that the shock load causes the movement of the medium particles along helical trajectories. Data on the types and velocities of shock waves are based on the analysis of the dynamic conditions for the compatibility of discontinuities; and these conditions are supplemented by the relations along the characteristic directions. We show that the leading front of the dynamic process in an undeformed previously medium is a plane-polarized shock wave which is simultaneously included in one of the families of characteristics. This property results in the constancy of shear direction on the plane-polarized shock wave. This condition makes it possible to significantly simplify the obtaining of approximate theoretical solutions for the near-front neighborhood of the shock wave. The two approximate solutions of the problem are presented. One of them bases on solving a system of evolutionary equations that was obtained using the matched asymptotic expansions method; and the second solution, on a version of the ray method.

Synthesis Control in the Systems with Distributed and Concentrated Parameters

They consider a controlled system with distributed and lumped parameters, the perturbed state of which is described by linear differential equations in partial and ordinary derivatives. The system consists of one distributed link with finite-dimensional links at its both ends (for example, the systems containing electrical circuits, sufficiently long elastic shafts, the pipelines in which it is necessary to take into account the distributed nature of the fluid flow, etc.). The paper studies the problem of a lumped optimal control development applied to finite-dimensional links and/or to the end of a distributed link that ensures the stable system operation. Two tasks are solved to develop such control. First, the method of Lyapunov functions is used to determine the set of controls that ensure the asymptotic stability ofthe closed-loop system; then, the Lagrange multiplier method is used on this set to develop an optimal control with the smallest value of the norm at each moment of time. At that, the original equations in private higher-order partial differential equations are represented as a system of evolutionary equations and first-order partial differential equations by introducing additional variables. The constraint equations are the equations without time derivatives. Amodified method of Lagrange multipliers is used to take such equations into account when calculating the derivative of the Lyapunov function in view of the system under consideration. The transition to partial differential equations of the first order, together with the development of ordinary differential equations in the standard Cauchy form, allows to construct the Lyapunov function as the sum of integral and ordinary quadratic forms constructively using specific equations, the sign-definiteness of which is checked by the Sylvester criterion, to ensure the stability of the closedloop system and obtain simpler control laws as the linear functions of phase coordinates. In the case of onedimensional distributed systems, these controls can be quite simply implemented in the form of lumped boundarycontrols that require measuring the system state only at the boundary points, which is of great practical importance. The results obtained in this work significantly expand the possibilities of Lyapunov functions method use when they solve applied problems of control synthesis in engineering objects with distributed and lumped parameters. As an example, we consider the problem of determining the minimum boost pressure of the fuel tank, which ensures stable operation of the heating furnace.

Asymptotic Behavior and Stability of a Stationary Boundary-Layer Solution to a Partially Dissipative System of Equations

A boundary value problem for a singularly perturbed partially dissipative system of two ordinary differential equations of the second and first order, respectively, is considered. An asymptotic expansion of its boundary-layer solution in a small parameter is constructed and justified. This solution is a stationary solution of the corresponding evolution system of equations with partial derivatives. The asymptotic stability of a stationary boundary-layer solution is proved, and its local basin of attraction is found.

Effect of a Magnetizable Surfactant on the Motion of a Liquid Film on the Horizontal Surface

The motion of a thin liquid film of viscous incompressible fluid on the horizontal surface in the presence of a magnetizable surfactant on the free boundary in the external inhomogeneous magnetic field is investigated. Surfactant diffusion along the free surface and the dependence of the surface tension on the magnetic field strength are taken into account. The system of evolutionary equations is derived in the lubricant approximation and steady-state film flows and their stability in the case of constant film thickness and constant surfactant number density are investigated with regard to the Marangoni effect.

Evolutionary Approach to Solve a Novel Time-Domain Cavity Problem

The problem of forced oscillations in a cavity filled with a dynamic plasma is solved in the time domain in compliance with the principle of causality. Interaction of the plasma and the cavity field is driven by the motion equation where the field is present in the Lorentz force standing at its right-hand side. The solution has a form of the modal expansions. Every term herein is a product of two multipliers. One is the vector element of the modal basis dependent on coordinates only. The basis is obtained as an eigenvector set of a self-adjoint operator specially selected from Maxwell’s equations. The operator is specified as a merger of the differential procedure $\nabla \times $ and the boundary conditions over perfectly conducting cavity surface. The basis elements are derived with their physical dimensions, i.e., volt per meter and ampere per meter. The other factor is a scalar dimension-free time-dependent modal amplitude. A system of evolutionary equations (i.e., with time derivative) for the modal amplitudes is derived and solved explicitly under the initial conditions. The problem is solved in the Hilbert space of real-valued functions of coordinates and time.

Nonexistence of a global solution of a mixed problem for a nonlinear system of equations with integral perturbation in a domain unbounded with respect to time variable

We study a mixed problem for a nonlinear connected evolution system of equations with integral perturbation in a domain unbounded with respect to the time variable. We prove that a global generalized solution of this problem does not exist.

The bound of supports of solutions for some classes of evolutionary systems and equations

We study the property of finiteness of a support for a system of evolutionary equations and for equations with nonstationary p-Laplacian and with convective terms. New bounds of the size of the support with regard for the convective terms are obtained.

On the existence of a generalized solution of a nonlinear evolution system of equations in a domain unbounded in time

We consider a mixed problem with Dirichlet homogeneous boundary conditions and nonzero initial conditions for a nonlinear coupled evolution system of equations in a domain unbounded in time. The conditions of existence of a generalized solution are obtained. It is shown that no solution of the problem exists at a negative initial value of the energy integral.

AN EXISTENCE AND UNIQUENESS RESULT FOR AN ELASTO-PIEZOELECTRIC PROBLEM WITH DAMAGE

The aim of this paper is to study the damage evolution in an elasto-piezoelectric body. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, and the piezoelectric effects are included into the model. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is then proved by using the theory of maximal monotone operators, the Schauder fixed-point theorem, and a comparison result.

An Elastic-viscoplastic Quasistatic Contact Problem: Existence and Uniqueness of a Weak Solution

In this paper, the contact between an elastic-viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, is also considered. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is established by using approximate problems, the pseudomonotone operators theory, Schauder fixed-point theorem and a comparison result.

Diffusion and creep in multi-component alloys with non-ideal sources and sinks for vacancies

Diffusion in single-phase multi-component crystalline systems can be characterized by three attributes: (i) the vacancy mechanism of diffusion for “slowly” diffusing substitutional components, (ii) the existence of non-ideal sources and sinks for vacancies and (iii) the “quick” diffusion of atoms of interstitial components. The first attribute leads to a constraint amongst fluxes of vacancies and substitutional components. The second attribute causes the non-uniform generation/annihilation of vacancies in the system being accompanied by dissipation and contributing to local swelling or shrinkage. The third attribute concerns the change of lattice constants due to the change of the concentration of interstitials and, therefore, also contributes to local swelling and shrinkage. The evolution equations for these systems have been derived from the conservation laws and the thermodynamic extremal principle. The equations represent an extension of evolution equations for multi-component systems with ideal or no sources and sinks for vacancies derived in a previous paper [Svoboda J, Fischer FD, Fratzl P, Kroupa A. Acta Mater 50 (2002) 1369]. Both diffusion and vacancy generation/annihilation are supposed to be proportional to driving forces of a chemical and mechanical nature. The appropriate evolution equations are derived, and it is shown that diffusion and vacancy generation or annihilation in isotropic materials depend on two different sets of driving forces and, thus, they can be considered as independent processes. Two relevant kinetic coefficients can be introduced for an isotropic body: the bulk and shear viscosities. However, the deformation state caused by the generation/annihilation of vacancies at grain boundaries is highly anisotropic.

Effects of solute mass transfer on the stability of capillary jets

The effects of mass transfer (e.g. via evaporation) of surface-active solutes on the hydrodynamic stability of capillary liquid jets are studied. A linear temporal stability analysis is carried out yielding evolution equations for systems satisfying general non- linear kinetic adsorption relations and accompanying surface constitutive equations. The discussion of the instability mechanism associated with the Marangoni effect clarifies that solute transfer into the jet is destabilizing whereas transfer in the opposite direction reduces instability. The general analysis is illustrated by a system satisfying Langmuir-type kinetic relations. Contrary to a clean system (i.e. in the absence of surfactants), reduced jet viscosity may lead to a substantial reduction in perturbation growth. Furthermore, the Marangoni effect gives rise to an overstability mechanism whereby perturbations whose dimensionless wavenumbers exceed unity grow with time through oscillations of increasing amplitude. The common diffusion-control approximation constitutes an upper bound which substantially overestimates the actual growth of perturbations. Considering solutes belonging to the homologous series of normal alcohols in water–air systems, the intermediate cases (e.g. hexanol–water–air which is ‘mixed-control’) are the most susceptible to Marangoni instability.

Self-similar spherically symmetric cosmological models withtwo scalar fields

We shall study the qualitative properties of the class of self-similarspherically symmetric models with two non-interacting scalar fieldswith exponential potentials. Numerous scalar fields, and particularlyscalar fields with exponential potentials, are known to arise in a varietyof theories of the fundamental interactions. In recent investigations ofthe cosmological implications of these theories a number of novelfeatures has been obtained, including the important assistedinflationary behaviour. In order to study this inflationaryscenario, we shall restrict our study here to the so-called`spatially' self-similar models. We derive the governing system ofevolution equations for the model, which is an autonomousfive-dimensional system of ordinary differential equations, in terms ofappropriately defined bounded variables. A monotonic function is obtainedand the local stability of the equilibria is determined, so that globalproperties of the physical phase space can be deduced. Numericalintegrations are also undertaken to complement the qualitative analysisand in order to investigate possible transient behaviour. From ourdynamical analysis we find that, depending on the parameters of themodels, in general the models begin expanding from a state with twoscalar fields which are massless, and subsequently either recollapse andagain evolve towards a model with two massless scalar fields or continueexpanding towards the assisted inflationary solution. Hence we find thatthere exists an open set of models that undergo assisted inflation.

Influence of a sureactant on the instability of a falling liquid film

The hydrodynamic instability of a film flow of a weak solution containing a soluble volatile surfactant is investigated. Diffusion of the surfactant in the liquid, its evaporation into the boundary gas medium, and the adsorption and desorption processes in the near-surface layer are taken into account. A system of evolutionary equations is derived and a steady-state solution film flow along a vertical surface and the stability of this flow are investigated for the simultaneous action of body and capillary forces and the Marangoni effect. Hydrodynamic and diffusion instability modes are detected and their properties are investigated for constant and variable surfactant concentration in the adsorbed sublayer.

A physical mechanism of large-scale structure generation in turbulent convection

The generation of large-scale structures during turbulent convection in a rotating layer of incompressible fluid heated by internal heat sources is considered. The results of a theoretical and experimental investigation of a physical mechanism of large-scale structure formation which operates under conditions of high-intensity small-scale turbulent convection and low boundary heat transfer are discussed. The theoretical investigation is based on a system of evolutionary equations obtained for the transverse space moments of the physical fields, which describes the motion in thin layers of rotating fluid. The stability of the solution of the mathematical model is studied using the small perturbation method. As a result, a condition of existence of longwave instability of the system and a criterion determining the threshold of its onset are obtained. The theoretical conclusions are confirmed by a series of experiments carried out on a laboratory model. The design of the laboratory apparatus and the experimental technique are described.

Constitutive equations in nonlinear damage mechanics

The classical method of solving problems concerning the delayed fracture of structural materials and elements is based on analysis of their stabilized stress strain state and the use of parametric relations which include the time or number of cycles to failure. Researchers have developed another approach employing the concept of internal variables along with a system of evolutionary equations and failure criteria. Entropy, internal energy, empirical temperature, internal time, strain hardening, and damage are the quantities most often chosen as the internal variables

Discontinuous solutions of one-dimensional hamiltonian systems

Systems of evolutionary equations in a class having many physical applications as models for nonlinear waves are investigated from the standpoint of hamiltonian theory. Although in most applications these equations are otherwise accountable as hyperbolic systems, they are not necessarily so and the present account bypasses this more familiar approach, gaining several novel advantages. Conservation laws are related through hamiltonian structure to symmetries of the evolutionary system. Then, on the basis of the established facts about conservation laws, various types of discontinuous solutions are classified systematically; and all possible shocks or other discontinuities are shown to have variational characterizations inherited from the hamiltonian description of continuously differentiable solutions. Focusing the variational properties established in general, simple algebraic representations are found for discontinuities of limitingly small strength. Finally, two examples are treated: the system of equations modelling long waves in liquid-filled elastic tubes, applying also to long water waves in horizontal channels of arbitrary cross-section; and the equations of one-dimensional gas dynamics.

On the evolution of systems with finite-range interactions

This paper is mainly devoted to establish a connection between the intuitive idea of finite-range interactions and the evolution equations for systems having only this kind of interaction. In it a different treatment from that made in a preceding paper is presented; but the new method is equivalent to the first one. Moreover, some intuitive arguments are given to see that the evolution equations obtained in the preceding paper for homogeneous as well as inhomogeneous systems subjected or not to external stationary fields are exact. An Appendix deals with verifying that the solution of Zwanzig’s equations for any distribution, function does not depend on the projection operators employed to select this distribution function from the density in phase space. In this Appendix the choice of the projection operators in our method is also justified.