Evolutionary Hemivariational Inequality Research Articles

Existence for fractional evolutionary inclusions involving nonlinear weakly continuous operators with applications

In this paper, our primary purpose is to settle the existence for a type of fractional-order evolutionary inclusions. By utilizing the Rothe method and the surjective result of weakly continuous operators, we show that the solution of the evolutionary equation exists under several kinds hypotheses on the data. Finally, the obtained results are applied to verify the existence of solutions for a time-fractional evolutionary hemivariational inequality, a time-fractional Navier–Stokes–Voigt equation and a quasistatic friction contact problem.

Existence and feedback control for a class of nonlinear evolutionary equations

In the paper we provide systematic approaches to study existence and feedback control for a new evolutionary equation involving pseudomonotone operators. We first establish several existence results for the evolutionary equation by exploiting the Rothe method and using a surjectivity result for multivalued pseudomonotone operators. Then we show the existence of feasible pairs for the feedback control problem by assuming some sufficient conditions. Moreover, the existence of solutions to an evolutionary hemivariational inequality is presented.

A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided.

Feedback control for nonlinear evolutionary equations with applications

In the paper we are concerned with the feedback control system governed by nonlinear evolutionary equations involving weakly continuous operators. By using the Rothe method and a surjectivity result for weakly continuous operators, we first present the solvability for the evolutionary equation. Then we show the existence of solutions to the feedback control system. We also consider an existence result for an optimal control problem. Moreover, we apply the main results to a class of differential variational inequalities, evolutionary hemivariational inequalities and the non-stationary Navier–Stokes–Voigt equation with a subgradient inclusion condition.

Continuous dependence and optimal control of a dynamic elastic-viscoplastic contact problem with non-monotone boundary conditions

<p style='text-indent:20px;'>In this paper, we consider continuous dependence and optimal control of a dynamic elastic-viscoplastic contact model with Clarke subdifferential boundary conditions. Since the constitutive law of elastic-viscoplastic materials has an implicit expression of the stress field, the weak form of the model is an evolutionary hemivariational inequality coupled with an integral equation. By providing some equivalent weak formulations, we prove the continuous dependence of the solution on external forces and initial conditions in the weak topologies. Finally, the existence of optimal solutions to a boundary optimal control problem is established.</p>

Rothe method and numerical analysis for a new class of fractional differential hemivariational inequality with an application

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion.

Numerical analysis of a dynamic viscoplastic contact problem

In this paper, we study a dynamic contact problem with Clarke subdifferential boundary conditions. The material is assumed to be viscoplastic which has an implicit expression of the stress field in constitutive law. The weak form of the model is governed by an evolutionary hemivariational inequality coupled with an integral equation. We study a fully discrete approximation scheme of the problem and bound the errors. Under appropriate solution regularity assumptions, optimal-order error estimates can be derived. Finally, a numerical example is also included to support our theoretical analysis. Particularly, it gives numerical evidence on the theoretically predicted optimal convergence order.

Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation

In this paper, we consider a new class of a dynamic system which couplesa nonstationary incompressible Navier-Stokes equation involving nonmonotone and multivalued boundary 条件 of subdifferential type,and a generalized quasilinear reaction-diffusion equation with the Neumann boundary condition.The variational formulation of the system is provided in the form of an evolutionary hemivariational inequality coupled with the nonlinear parabolic equation. The main result concerning with the existence of global weak solutions to the system is proved.Firstly, the backward Euler technique and the feedback iterative method are used to develop a temporally semi-discrete approximate 问题. Next, invoking a surjectivity result for weakly-weakly upper semicontinuous multivalued operators, the existence and $a$ $priori$ estimates for the 解 to the semi-discrete approximate 问题 are established. Finally, using a limiting procedure for solutions to the approximate 问题, a 定理 on the existence of weak solutions to the system is obtained.

Feedback control for non-stationary 3D Navier–Stokes–Voigt equations

The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.

Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

In this paper, an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem, we provide the optimal error estimate.

Evolutionary Boussinesq model with nonmonotone friction and heat flux boundary conditions

In this paper we prove the existence and regularity of a solution to a two-dimensional system of evolutionary hemivariational inequalities which describes the Boussinesq model with nonmonotone friction and heat flux. We use the time retardation and regularization technique, combined with a regularized Galerkin method, and recent results from the theory of hemivariational inequalities.

Analysis and control of differential inclusions with anti-periodic conditions

We deal with the optimal control of systems driven by differential inclusions with anti-periodic conditions in Banach spaces. The operators in the differential inclusions, describing the control system, all depend on control variables. By a solution of the control system we mean a ‘trajectory-control’ pair. We first prove that, for fixed control variables, the set of solutions is non-empty for the corresponding differential inclusion with anti-periodic conditions. We then show the functional-topological properties of the set of solutions. Finally, we investigate the existence of solutions to the optimal control problem. These abstract results can be used to study optimal control systems of evolutionary hemi-variational inequalities.

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.

An evolution problem in nonsmooth elasto-viscoplasticity

In this note we summarize some of our recent results on the mathematical modeling of contact problems. We concentrate on a dynamical model which describes the frictional contact between a deformable body and a foundation. The material behavior is described with a nonlinear elastic–viscoplastic constitutive law and the frictional contact is modeled with multivalued subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. We prove the existence and the regularity properties of a unique weak solution to the model.

Comparison Principle For Quasilinear Parabolic Inclusions with Clarke’s Gradient

Abstract We prove existence and comparison results for quasilinear parabolic inclusions with Clarke’s generalized gradient via appropriately defined sub-supersolution only assuming a local Lq-boundedness condition on Clarke’s gradient. The developed comparison principle allows us to show the equivalence of the considered parabolic inclusion with an associated evolutionary hemivariational inequality.

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.

Analysis of a dynamic Elastic-Viscoplastic contact problem with friction

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with an elastic-viscoplastic constitutive law and the frictional contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators and a fixed point theorem.