Nonlinear Parabolic Variational Inequalities Research Articles

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

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

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>

Numerical analysis and simulations of a frictional contact problem with damage and memory

<p style='text-indent:20px;'>In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.</p>

Unsteady non-Newtonian fluid flows with boundary conditions of friction type: The case of shear thickening fluids

Motivated by applications to industrial processes like lubrication or extrusion/ injection we consider in this paper non-stationary flow problems for general incompressible dilatant (shear thickening) fluids. The conservation of mass and momentum lead to a p-Laplacian unsteady Stokes system where the real parameter p is greater than 2. Such fluids undergo complex slip fluid–solid interface laws of friction type which may be described by a subdifferential boundary condition. Hence the fluid velocity and pressure satisfy a non-linear parabolic variational inequality and belong to Banach spaces depending on p. We prove the existence of a solution by using a fixed point technique combined with compactness and monotonicity arguments.

A topological degree theory for constrained problems with compact perturbations and application to nonlinear parabolic problem

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊇ D ( T ) → 2 X ∗ be maximal monotone, S : X → 2 X ∗ be bounded of type ( S + ) and C : X ⊇ D ( C ) → 2 X ∗ be compact with D ( T ) ⊆ D ( C ) . A degree theory is developed for operators of the type T + S + C . In general, the operator S + C : D ( C ) → 2 X ∗ is not necessarily an operator of type ( S + ) because D ( C ) is neither convex nor closed. Thus developing a degree theory for these class of operators is relevant as it generalizes the theories for operators of the type T + S . A new existence theorem is given for the solvability of the constrained problem T u + S u + C u ∋ 0 u ∈ D ( T ) . The case C = { 0 } gives those theories and existence theorems for operators of the type T + S . An application on the existence of solution for a nonlinear parabolic variational inequality problem is provided.

Existence and Uniqueness of Viscosity Solutions for Nonlinear Variational Inequalities Associated with Mixed Control

The author investigates the nonlinear parabolic variational inequality derived from the mixed stochastic control problem on finite horizon. Supposing that some sufficiently smooth conditions hold, by the dynamic programming principle, the author builds the Hamilton-Jacobi-Bellman (HJB for short) variational inequality for the value function. The author also proves that the value function is the unique viscosity solution of the HJB variational inequality and gives an application to the quasi-variational inequality.

Numerical analysis and simulation of an adhesive contact problem with damage and long memory

<p style='text-indent:20px;'>This paper studies an adhesive contact model which also takes into account the damage and long memory term. The deformable body is composed of a viscoelastic material and the process is taken as quasistatic. The damage of the material caused by the compression or the tension is involved in the constitutive law and the damage function is modelled through a nonlinear parabolic inclusion. Meanwhile, the adhesion process is modelled by a bonding field on the contact surface while the contact is described by a nonmonotone normal compliance condition. The variational formulation of the model is governed by a coupled system which consists of a history-dependent hemivariational inequality for the displacement field, a nonlinear parabolic variational inequality for the damage field and an ordinary differential equation for the adhesion field. We first consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived. At the end of this paper, {we report some two-dimensional numerical simulation results} for the contact problem in order to provide numerical evidence of the theoretical results.

Numerical analysis of a bone remodeling model with damage

Bone tissue is a material with a complex structure and mechanical properties. Repetitive loads or diseases can cause microfractures to appear in the bone tissue, which results in a deterioration of its mechanical properties. On the other hand, bone is a constantly evolving tissue, adapting its density to the loading conditions it is subjected to. In this work, we study, from the numerical point of view, a strain-adaptive bone remodeling model coupled with a damage model. The variational problem is written as a coupled system consisting of a nonlinear variational equation for the displacement field and nonlinear parabolic variational inequalities for the apparent density and damage. Then, fully discrete approximations are introduced, using the finite element method and a hybrid combination of Euler schemes. A priori error estimates are proved under adequate regularity conditions, and the linear convergence of the algorithm is deduced. Finally, some one- and two-dimensional numerical simulations of test examples are presented, to demonstrate the accuracy of the approximations and the behavior of the solutions.

Rothe's method for nonlinear parabolic variational inequalities in noncylindrical domains

In this paper, a nonlinear parabolic variational inequality in noncylindrical domain is considered. Using extended Rothe’s method recently achieved in [10] an approximate solution is constructed. Existence and uniqueness results are proved. Also, we present some further results and comments related to the main result.

Unsteady 3D-Navier–Stokes system with Tresca’s friction law

We consider unsteady incompressible 3D fluid flow with nonconstant viscosity subjected to nonlinear boundary conditions of Tresca’s type: when the modulus of the shear stress at the boundary is smaller than a given threshold, the classical no-slip condition holds; otherwise, the modulus of the shear stress at the boundary equals the given threshold and the fluid is allowed to slip. The problem is thus described by a nonlinear parabolic variational inequality. We construct a sequence of approximate solutions by using both a regularization of the free boundary condition due to friction and a penalty method, reminiscent of the “incompressibility limit” of compressible fluids, allowing us to get a better insight into the links between the fluid velocity and pressure fields. Then we pass to the limit with compactness arguments to obtain a solution to our original problem.

A note on strongly nonlinear parabolic variational inequalities

Abstract We prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

On the convergence of an explicit difference scheme for evolution variational inequalities with nonlocal space operator

A nonlinear parabolic variational inequality with nonlocal space operator monotone with respect to gradient is considered. An explicit difference scheme with respect to the space operator and an implicit difference scheme with respect to the penalty operator are constructed by using the penalty method and the method of integral identities. Stability conditions for the constructed difference scheme are obtained. A convergence theorem with minimal assumptions on the smoothness of the initial data is proved.

Ergodicity of Robust Switching Control and Nonlinear System of Quasi-Variational Inequalities

We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments and in particular a dual randomized game representation for the solution to the system of variational inequalities.

Global solvability of a model for grain boundary motion with constraint

We consider a model for grain boundary motion with constraint. In composite material science it is very important to investigate the grain boundary formation and its dynamics. In this paper we study a phase-filed model of grain boundaries, which is a modified version of the one proposed by R. Kobayashi, J.A. Warren and W.C. Carter [18]. The model is described as a system of a nonlinear parabolic partial differential equation and a nonlinear parabolic variational inequality. The main objective of this paper is to show the global existence of a solution for our model, employing some subdifferential techniques in the convex analysis.

Numerical analysis of a strain-adaptive bone remodelling problem

In this paper we study from the numerical point of view a strain-adaptive bone remodelling that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. The variational problem is written as a coupled system of a nonlinear variational equation for the displacement field and a nonlinear parabolic variational inequality for the apparent density. Then, fully discrete approximations are provided by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are proved, from which, under adequate regularity conditions, the linear convergence of the algorithm is deduced. Finally, some numerical simulations involving one- and two-dimensional test examples are presented to demonstrate the accuracy of the approximation and the behaviour of the solution.

Higher-order parabolic variational inequality in unbounded domains

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity.

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.

On nonlinear parabolic variational inequalities containing nonlocal terms

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered: $$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$ (1) where V denotes a closed linear subspace of the Sobolev-space W1,p(Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.

On a few properties of solutions of nonlinear parabolic variational inequalities

On a few properties of solutions of nonlinear parabolic variational inequalities

On the box method for a non-local parabolic variational inequality

In this paper we study a box scheme (or finite volume element method) for anon-local nonlinear parabolic variational inequality arising in the studyof thermistor problems. Under some assumptions onthe data and regularity of the solution, optimal error estimates in the$H^1$-norm are attained.