Evolutionary Variational Inequalities Research Articles

Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

On the Mathematical Analysis of Thick Fluids

In chemical engineering models, shear-thickening or dilatant fluids converge in the limit case to a class of incompressible fluids with a maximum admissible shear rate, the so-called thick fluids. These non-Newtonian fluids can be obtained, in particular, as the power limit of the Ostwald–de Waele fluids, and can be described as a new class of evolution variational inequalities, in which the shear rate is bounded by a positive constant or, more generally, by a bounded positive function. It is established the existence, uniqueness, and the continuous dependence of solutions to this general class of thick fluids with variable threshold on the absolute value of the deformation rate tensor, the solutions of which belong to a time dependent convex set. For sufficiently large viscosity, the asymptotic stabilization toward a unique steady state is also proved.

On the Existence of Strongly Continuous Physical Solutions for Classes of Autonomous Evolutionary Variational Inequalities

UDC 517.9 Abstract. The concept of a physical solution (on a finite time interval) is introduced on the basis of natural energy equalities and continuous dependence of state functions on time variables in the phase space for classes of autonomous evolutionary variational inequalities on convex cones with nonlinear nonmonotone (in the general case) mappings. For approximate searching for such solutions, the classical penalty method is used. For the solutions obtained, this article substantiates the possibility of a global description of the behavior of such systems in the natural phase space with respect to the topology of strong convergence by finite algorithms up to an arbitrary small parameter.

A two obstacles coupled problem

We consider a system of an evolutionary variational inequality of two obstacles type, depending on the temperature, coupled with the heat equation. We prove existence of solution of this system and we present examples that motivated this work. In particular, with additional assumptions on the data, we prove that solutions of this problem are also solutions of a similar problem where the convex set is of gradient constraint type (that depends on the temperature), improving a previous result.

Deriving amplitude equations via evolutionary $\Gamma$-convergence

We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary $\Gamma$-convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show $\Gamma$-convergence of the associated energies in suitable function spaces.  &nbsp The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in $L^2$, while for the case of a quadratic nonlinearity we need to impose weak convergence in $H^1$. However, we do not need well-preparedness of the initial conditions.

Evolutionary variational inequality with long-term memory and applications to economic networks

The aim of this paper is to present the relation between an evolutionary variational inequality with long-term memory and Lagrange multipliers. More precisely, we study the oligopolistic market equilibrium problem in which the profit function depends also on previous events of the market by means of a long-term memory which takes into account the previous states of the equilibrium. Moreover, thanks to the variational formulation, we are able to show existence and regularity results for equilibrium solutions. Then, we apply the infinite dimensional duality theory through which we obtain the existence of Lagrange multipliers which are great utility in order to understand the behaviour of the market. Finally, an example is provided, which allows to analyse the influence of the long-term memory on the equilibrium solution.

Regularity for nonlinear evolution variational inequalities with delay terms

In this paper, we deal with the regularity and a variation of constant formula for solutions of the nonlinear differential equation with delay nonlinear terms governed by the variational inequality in Hilbert spaces. Without the conditions of the uniform boundedness of the nonlinear terms and the compactness of the principal operators, we obtain the wellposedness and the norm estimate of the given equation by converting the problem into the contraction mapping principle on state space.MSC:35K85, 35F25.

Inverse Variational Inequality Approach and Applications

It is well known that a dynamic oligopolistic market equilibrium problem can be studied as an evolutionary variational inequality and this problem is approached as a problem of profit optimization for the firms. On the contrary, in this article, with the help of an inverse variational formulation, the behavior of control policies for an oligopolistic market equilibrium problem, whose aim is to regulate the exportation through the adjustment of taxes on the firms, is studied. This is considered as a policymaker optimization problem. More precisely, a definition of equilibrium for the firms by using the Lagrange multipliers is provided together to the optimal regulatory tax definition. Moreover an existence result is given and, at last, a numerical example is analyzed.

On the solvability of an evolution variational inequality with a nonlocal space operator

By using semidiscretization and penalty methods, we prove the existence of a generalized solution of a variational inequality of parabolic type with a space operator monotone in the gradient and depending on an integral characteristic of the solution with respect to the space variables.

Dimensional contraction via Markov transportation distance

It is now well known that curvature conditions à la Bakry–Émery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strengthening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove, in the setting of a compact Riemannian manifold, that this Markov transportation distance satisfies the same properties as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and evolution variational inequalities.

Hemivariational Inequality Approach to Evolutionary Constrained Problems on Star-Shaped Sets

In this paper, we consider a nonconvex evolutionary constrained problem for a star-shaped set. The problem is a generalization of the classical evolution variational inequality of parabolic type. We provide an existence result; the proof is based on the hemivariational inequality approach, a surjectivity theorem for multivalued pseudomonotone operators in reflexive Banach spaces, and a penalization method. The admissible set of constraints is closed and star-shaped with respect to a certain ball; this allows one to use a discontinuity property of the generalized Clarke subdifferential of the distance function. An application of our results to a heat conduction problem with nonconvex constraints is provided.

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sem Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$ C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$ C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$ C ( t ) has a continuous variation for every $$t\in [0,T]$$ t ? [ 0 , T ] with $$C(0)$$ C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Kreja?i in Eur J Appl Math 2:281---292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179---186, 1973 and Cornet in J. Math. Anal. Appl. 96:130---147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.

The equilibrium of closed-loop supply chain supernetwork with time-dependent parameters

In this paper, we develop a closed-loop supply chain supernetwork model including suppliers, manufacturers, retailers and consumers at demand market, in which the demand for product is seasonal (t), and the sensitivity (w) of demand to price is another key factor which effects consumers’ demand. Moreover, the manufacturers invest the reverse distribution channel for incenting consumers to return more used products. Based on the Evolutionary Variational Inequalities (EVI) theory and Projected Dynamical Systems (PDS), the equilibrium condition of closed-loop supply chain is formulated, and the model is verified reasonably by numerical example.

Stochastic variational inequalities associated with elasto-plastic torsion

This paper is concerned with an initial value problem for a stochastic variational inequality associated with elasto-plastic torsion. Our goal is to establish the existence and uniqueness of a solution. The stochastic problem is reduced to essentially a deterministic problem, which is not covered by existing results on evolution variational inequalities. We propose a definition of a solution in the same spirit as for weak solutions of partial differential equations, and derive some basic consequences of our definition. Based on these results, we can prove the existence and uniqueness of a solution to the stochastic problem.

An Inverse Problem for the Dynamic Oligopolistic Market Equilibrium Problem in Presence of Excesses

The theory of evolutionary variational inequalities fits very well in a lot of optimization problems, like for example the dynamic oligopolistic market equilibrium problem that we describe in this paper. But here, the authors analyze the behavior of control policies whose aim is to regulate the exportation through the adjustments of supply taxes or incentives on the firms. This is considered as a policy-maker optimization problem. This aspect is studied with the help of an inverse evolutionary variational formulation. Moreover, a characterization of the inverse variational inequality with an appropriate evolutionary variational inequality is given. Unlike the paper (Barbagallo & Mauro, 2013), here the authors explore the possibility of presence of production and demand excesses. Moreover, we provide a definition of equilibrium for the firms by using the infinite dimensional duality theory and later we define an optimal regulatory tax. Finally, we present a numerical scheme in order to compute the equilibrium solution of the inverse variational inequality that models the policy-maker's point of view for the dynamic oligopolistic market equilibrium problem in presence of excesses, then we provide a numerical example.

An Approach to Nonlinear Viscoelasticity via Metric Gradient Flows

We formulate quasistatic nonlinear finite-strain viscoelasticity of rate type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can be obtained only by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous limit in a specific case.

Dynamic Traffic Network Equilibrium System with Elastic Demands

We discuss the dynamic traffic network equilibrium system problem with elastic demands by way of a quasi-variational inequality.We introduce the equilibrium definition with elastic demands based on Wardrops principles when there are some internal relationships between different kinds of goods which transported through the same traffic network. Moreover, we also prove that the equilibrium conditions of this problem with elastic demands can be equivalently expressed as a system of evolutionary variational inequalities. By using the fixed point theory and projected dynamic system theory, we get the existence and uniqueness of the solution for this equilibrium problem with elastic demands. Finally, some numerical examples are given to illustrate our results.

Sandpiles and superconductors: nonconforming linear finite element approximations for mixed formulations of quasi-variational inequalities

Similar evolutionary variational and quasi-variational inequalities with gradient constraints arise in the modelling of growing sandpiles and type-II superconductors. Recently, mixed formulations of these inequalities were used for establishing existence results in the quasi-variational inequality case. Such formulations, and this is an additional advantage, made it possible to determine numerically not only the primal variables, e.g., the evolving sand surface and the magnetic field for sandpiles and superconductors, respectively, but also the dual variables, the sand flux and the electric field. Numerical approximations of these mixed formulations in previous studies employed the Raviart–Thomas element of the lowest order. Here we introduce simpler numerical approximations of these mixed formulations based on the nonconforming linear finite element. We prove (subsequence) convergence of these approximations and illustrate their effectiveness by numerical experiments.

Internal and subspace correction approximations of implicit variational inequalities

The aim of this paper is to study the existence of solutions and some approximations for a class of implicit evolution variational inequalities that represents a generalization of several quasistatic contact problems in elasticity. Using appropriate estimates for the incremental solutions, the existence of a continuous solution and convergence results are proved for some corresponding internal approximation and backward difference scheme. To solve the fully discrete problems, general additive subspace correction algorithms are considered, for which global convergence is proved and some error estimates are established.

Variational and numerical analysis of a quasistatic viscoelastic contact problem with normal compliance and friction

In this paper, a class of generalized evolution variational inequalities arising in quasistatic friction contact problem for viscoelastic materials is introduced and studied. Under some suitable assumptions, we obtain an existence and uniqueness theorem of the solution for the generalized evolution variational inequalities by using Banach’s fixed point theorem. Moreover, we study two numerical approximation schemes of the problem: semidiscrete scheme and fully discrete scheme. For both schemes, we prove the existence of the solution and derive the error estimations.