scholarly journals On the well-posedness of differential quasi-variational-hemivariational inequalities

2020 ◽  
Vol 18 (1) ◽  
pp. 540-551 ◽  
Author(s):  
Jinxia Cen ◽  
Chao Min ◽  
Van Thien Nguyen ◽  
Guo-ji Tang
Keyword(s):  
Hilbert Spaces ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Well Posed

Abstract The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.

scholarly journals Well-posedness for generalized variational-hemivariational inequalities with perturbations in reflexive banach spaces

2017 ◽  
Vol 48 (4) ◽  
pp. 345-364 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Yung-Yih Lur ◽  
Ching-Feng Wen
Keyword(s):  
Banach Spaces ◽  
Minimization Problem ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Reflexive Banach Spaces ◽  
Mild Conditions ◽  
Well Posed

In this paper, we consider an extension of well-posedness for a minimization problem to a class of generalized variational-hemivariational inequalities with perturbations in reflexive Banach spaces. We establish some metric characterizations for the $\alpha$-well-posed generalized variational-hemivariational inequality and give some conditions under which the generalized variational-hemivariational inequality is strongly $\alpha$-well-posed in the generalized sense. Under some mild conditions, we also prove the equivalence between the $\alpha$-well-posedness of the generalized variational-hemivariational inequality and the $\alpha$-well-posedness of the corresponding inclusion problem.

scholarly journals Well-Posedness for a Class of Strongly Mixed Variational-Hemivariational Inequalities with Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Minimization Problem ◽  
Hemivariational Inequality ◽  
The Other ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Mild Conditions ◽  
Other Hand ◽  
Well Posed ◽  
Special Case

The concept of well-posedness for a minimization problem is extended to develop the concept of well-posedness for a class of strongly mixed variational-hemivariational inequalities with perturbations which includes as a special case the class of variational-hemivariational inequalities with perturbations. We establish some metric characterizations for the well-posed strongly mixed variational-hemivariational inequality and give some conditions under which the strongly mixed variational-hemivariational inequality is strongly well-posed in the generalized sense. On the other hand, it is also proven that under some mild conditions there holds the equivalence between the well posedness for a strongly mixed variational-hemivariational inequality and the well-posedness for the corresponding inclusion problem.

scholarly journals Well-posedness by perturbations of variational-hemivariational inequalities with perturbations

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 881-895 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Himanshu Gupta ◽  
Ching-Feng Wen
Keyword(s):  
Variational Inequalities ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Mild Conditions ◽  
Finite Dimensional ◽  
Mixed Variational Inequalities ◽  
Well Posed ◽  
Special Case

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a class of variational-hemivariational inequalities with perturbations in Banach spaces, which includes as a special case the class of mixed variational inequalities. Under very mild conditions, we establish some metric characterizations for the well-posed variational-hemivariational inequality, and show that the well-posedness by perturbations of a variational-hemivariational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Furthermore, in the setting of finite-dimensional spaces we also derive some conditions under which the variational-hemivariational inequality is strongly generalized well-posed-like by perturbations.

scholarly journals Partial differential hemivariational inequalities

2018 ◽  
Vol 7 (4) ◽  
pp. 571-586 ◽  
Author(s):  
Zhenhai Liu ◽  
Shengda Zeng ◽  
Dumitru Motreanu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Mild Solutions ◽  
Upper Semicontinuity ◽  
Solution Set ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Partial Differential ◽  
New Class

AbstractThe aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.

scholarly journals Well-posedness analysis of a stationary Navier–Stokes hemivariational inequality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min Ling ◽  
Weimin Han
Keyword(s):  
Stokes Equations ◽  
Normal Component ◽  
Numerical Algorithms ◽  
Hemivariational Inequality ◽  
Navier Stokes ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Solution Existence And Uniqueness ◽  
Stationary Navier Stokes Equations

AbstractThis paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

scholarly journals Well-Posedness by Perturbations for Variational-Hemivariational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Shu Lv ◽  
Yi-bin Xiao ◽  
Zhi-bin Liu ◽  
Xue-song Li
Keyword(s):  
Optimization Problem ◽  
Hemivariational Inequality ◽  
Inclusion Problem ◽  
Hemivariational Inequalities ◽  
Well Posedness

We generalize the concept of well-posedness by perturbations for optimization problem to a class of variational-hemivariational inequalities. We establish some metric characterizations of the well-posedness by perturbations for the variational-hemivariational inequality and prove their equivalence between the well-posedness by perturbations for the variational-hemivariational inequality and the well-posedness by perturbations for the corresponding inclusion problem.

scholarly journals Differential variational–hemivariational inequalities: existence, uniqueness, stability, and convergence

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Guo-ji Tang ◽  
Jinxia Cen ◽  
Van Thien Nguyen ◽  
Shengda Zeng
Keyword(s):  
Nonlinear Evolution Equation ◽  
Original Problem ◽  
Nonlinear Evolution ◽  
Convergence Result ◽  
Hemivariational Inequality ◽  
Stability And Convergence ◽  
Hemivariational Inequalities ◽  
Well Posedness ◽  
Kkm Principle ◽  
Stability Of The Solution

AbstractThe goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

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