Parabolic Variational Inequalities Research Articles

Accuracy of an implicit scheme for the finite element method with a penalty for a nonlocal parabolic obstacle problem

In order to solve a parabolic variational inequality with a nonlocal spatial operator and a one-sided constraint on the solution, a numerical method based on the penalty method, finite elements, and the implicit Euler scheme is proposed and studied. Optimal estimates for the accuracy of the approximate solution in the energy norm are obtained.

Periodic solutions to a class of quasi-variational evolution equations

We study a time-periodicity problem for a class of abstract nonlinear evolution equations associated with subdifferential operators depending on both time and the unknown. Assuming time-periodicity for the subdifferential operator, we prove the existence of a periodic solution using the abstract theory of time-dependent subdifferential evolution equations and its generalization. We provide application examples for quasilinear parabolic variational inequalities with time-periodic constraints of non-local time-delay or memory type depending on the unknown.

A generalized penalty method for a new class of differential inequality system

The primary objective of this paper is to study a nonlinear system involving a parabolic variational inequality, a history-dependent hemivariational inequality and a differential equation constraints in a Banach space. First, we derive a unique solvability theorem for such problem under some mild hypotheses. Second, we construct a penalized problem for such nonlinear system, and show the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the sequence of approximate solution to the solution of the original system when the penalty parameter converges to zero. Finally, these results are applied to a quasistatic elastic frictional contact problem with heat equation with memory, and damage.

Existence of solutions for parabolic variational inequalities

Existence of solutions for parabolic variational inequalities

Frictional Contact Problem With Wear for Thermo-Viscoelastic Materials With Damage

We consider a mathematical model which describes a dynamic frictional contact problem for thermo-viscoelastic materials with long memory and damage. The contact is modeled by the normal compliance condition and wear between surfaces are taken into account. We establish a variational formulation for the model and prove the existence and uniqueness of the weak solution. The proof is based on arguments of hyperbolic nonlinear differential equations, parabolic variational inequalities and Banach fixed point.

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains

Poroelastic problem of a non-penetrating crack with cohesive contact for fluid-driven fracture

A new class of unilaterally constrained problems for fully coupled poroelastic models stemming from hydraulic fracturing is introduced and studied with respect to its well-posedness. The poroelastic medium contains a fluid-driven crack, which is subjected to non-penetration conditions and cohesion forces between the crack faces. Existence of solution for the governing elliptic–parabolic variational inequality under the unilateral constraint with a small cohesion is established using the incremental approximation based on Rothe’s semi-discretization in time.

A Simple Line-Element Model for Three-Dimensional Analysis of Steady Free Surface Flow through Porous Media

Considering the fact that only pores can transport water, pores in the homogeneous control volume are conceptualized as a three-dimensional orthogonal network of line elements, which is in contrast to the continuum hypothesis in traditional numerical approaches. The related flow velocity, hydraulic conductivity and continuity equation equivalent to the continuum model are formulated based on the principle of flow balance. Subsequently, the unified form for flow velocity and continuity equation is established based on the local coordinate system, and a finite line-element method is developed, in which three-dimensional steady free surface flow is reduced to one-dimensional form, and the numerical difficulty is greatly decreased. The proposed line-element model is validated by the good agreements of free surface locations with other methods through steady flow in a rectangular dam and a right trapezoidal dam, respectively. It is found that the proposed line-element model is not heavily dependent on the mesh size and penalty parameter. Steady free surface flow on the left bank abutment slope of the Kajiwa Dam in Southwestern China is further evaluated, and a parabolic variational inequality algorithm based on the continuum model is also employed for comparison. The consistent results indicate that the proposed line-element model can capture the steady free surface flow behavior as well as the continuum-based method. Moreover, the proposed line-element model can rapidly achieve accurate solutions whether for simple examples or for complicated engineering applications.

A free boundary problem for a flexible loan based on the borrower asset

In this paper, a restricted free boundary problem model for the remaining loan value under a flexible repayment method is proposed. The model is formulated as a weak coupling problem between parabolic variational inequality and partial differential equation. The existence and uniqueness of the solution are obtained. Some properties of the free boundary, such as boundedness, monotonicity and smoothness, are also studied. Especially, the relationship between the free boundary and its lower bound threshold is discussed. As the expected clear off time of the loan cannot be analytically expressed, numerical calculation is shown. Some numerical results for the free boundary are also presented.

Existence, uniqueness, and stabilization results for parabolic variational inequalities

In this paper, we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau–Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.

Lewy-Stampacchia inequality for noncoercive parabolic obstacle problems

<abstract><p>We investigate the obstacle problem for a class of nonlinear and noncoercive parabolic variational inequalities whose model is a Leray–Lions type operator having singularities in the coefficients of the lower order terms. We prove the existence of a solution to the obstacle problem satisfying a Lewy-Stampacchia type inequality.</p></abstract>

Regularity of weak solutions to a class of fourth order parabolic variational inequality problems arising from swap option pricing

<abstract><p>This article concerns the regularity of weak solutions for a variational inequality problem constructed by a fourth-order parabolic operator which has received much attention recently. We first consider the internal regular estimate of weak solutions using the difference type test function. Then, the near edge regularity and global regularity of weak solutions are analyzed by using the finite cover principle. Since the quadratic gradient of the weak solution does not satisfy the conditions for a test function, we have constructed a test function using a spatial difference operator to complete the proof of regularity. The results show that the weak solution has a second order regularity and an $ {L^\infty }(0, T;{H^2}(\Omega)) $ estimation independent of the lower order norm of the weak one.</p></abstract>

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

Backward multivalued McKean-Vlasov SDEs and associated variational inequalities

The work concerns a type of backward multivalued McKean-Vlasov stochastic differential equations. First, we prove the existence and uniqueness of solutions for backward multivalued McKean-Vlasov stochastic differential equation. Then, it is presented that the first components of their solutions depend continuously on the terminal values. Finally, we give a probabilistic interpretation for viscosity solutions of nonlocal quasi-linear parabolic variational inequalities.

Study of weak solutions for degenerate parabolic inequalities with nonstandard conditions

In this paper, we study the degenerate parabolic variational inequalities in a bounded domain. By solving a series of penalty problems, the existence and uniqueness of the solutions in the weak sense are proved by the energy method and a limit process.

Convergence analysis of a symmetric dual-wind discontinuous Galerkin method for a parabolic variational inequality

This paper investigates a symmetric dual-wind discontinuous Galerkin (DG) method for solving parabolic variational inequalities. By employing a symmetric dual-wind DG discretization in space and a backward Euler discretization in time, we propose a fully discrete scheme to solve a time-dependent obstacle problem. Under reasonable regularity assumptions on the exact solution, we prove the convergence of numerical solutions with rates in the L∞(L2) and L2(H1)-like energy errors by introducing a new interpolation operator which is a combination of the standard interpolation operator and a positive-preserving interpolation operator. Numerical experiments are provided to validate the effectiveness of the proposed method.

A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem

The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.

Penalty method for a class of differential nonlinear system arising in contact mechanics

The main goal of this paper is to study a class of differential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We first construct a penalized problem for such a nonlinear system and then derive the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the obtained approximating sequence to the solution of the original nonlinear system when the penalty parameter converges to zero. Finally, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics. First part title: IntroductionSecond part title: PreliminariesThird part title: Convergence result for (1.1)Fourth part title: An application

Study of Weak Solutions for a Class of Degenerate Parabolic Variational Inequalities with Variable Exponent

The authors of this text study weak solutions to the variational inequalities with degenerate parabolic operators and symmetric structure in Sobolev spaces with variable exponent. The existence and uniqueness of the solutions are treated by using the penalty method and the reduction method in the weak sense. The authors also discuss the nonexistence and long-time behavior of solutions.

Averaging principle for stochastic variational inequalities with application to PDEs with nonlinear Neumann conditions

Stochastic variational inequalities have been widely used in various areas. In this paper we establish averaging principles for a separated time-scale system of fully coupled stochastic system characterized by stochastic variational inequalities. Under non-Lipschitz continuous conditions, we show that the classical weak convergence result holds for this type of stochastic systems. Strong convergence is also studied for the cases when the diffusion coefficients of the slow motions do not depend on the fast motion components. As an application, we study the homogenization of generalized backward SDEs and semilinear parabolic variational inequalities with nonlinear Neumann boundary conditions.