Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces

Migórski, Stanisław; Pączka, Dariusz

doi:10.1007/978-3-030-15242-0_1

Cited by 2 publications

(4 citation statements)

References 55 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By Lemma 2, it follows that d 0 (u λ ; u 0 − u λ ) ≤ 0, so we can skip the last term in estimate (23). We deduce that u λ V ≤ M with M > 0 independent of λ > 0.…”

Section: Proof Of Theoremmentioning

confidence: 85%

“…can be found in classical monographs [8,9,16], and in two recent books [17,18], and the references therein. A unified method, based on the hemivariational inequality formulation, to study contact problems of viscoelasticity is given in [19], the abstract elliptic variational-hemivariational inequalities in reflexive Banach spaces with applications can be found in [1], and the variational-hemivariational inqualities which model fluid flow in mechanics were treated in [20,21] and very recently in [22,23]. Other recent developments on variational methods in the study of existence and multiplicity of solutions, see [24,25].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Migórski

Fengzhen

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

In this paper, we study a class of constrained variational–hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational–hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational–hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

show abstract

“…By Lemma 2, it follows that d 0 (u λ ; u 0 − u λ ) ≤ 0, so we can skip the last term in estimate (23). We deduce that u λ V ≤ M with M > 0 independent of λ > 0.…”

Section: Proof Of Theoremmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Migórski

Fengzhen

2020

Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the stationary case, the problem (1.1)-(1.3) with nonconvex superpotentials j was considered by Migorski and Ochal [27] Migroski [26], for non-newtonian case see [14] . In Orcliz spaces, hemivariational inequalities for Newtonian and Non-newtonian Navier-Stokes equations has been recently studied in [25], [24]. Hemivariational inequalities for generalized Newtonian fluids are recently extensively studied see [13] and references therein, see also [23] for evolutionary Oseen model for generalized Newtonian fluid.…”

Section: Introductionmentioning

confidence: 99%

“…For an equilibrium problem approach to hemivariational inequalities for Navier-Stokes equations we refer to [1] and [4]. For different aspects about nonsmooth optimization in the context of Navier-Stokes system we refer to [16,17,19,20,33,31,32,33,34,50] The goal of this paper is threefold. We aim to (1) show the existence of weak solutions to the hemivariational inequality corresponding to the problem (1.1)-(1.3), (2) prove a dependence result of solutions with respect to the hemivariational part and to the density of the external forces, (3) formulate and study the distributed parameter optimal control where the control is represented by the density of the external forces.…”

Section: Introductionmentioning

confidence: 99%

Hemivariational Inequality for Navier–Stokes Equations: Existence, Dependence, and Optimal Control

Mahdioui

Aadi

Akhlil

2020

Bull. Iran. Math. Soc.

View full text Add to dashboard Cite

In this paper we study existence, dependence and optimal control results concerning solutions to a class of hemivariational inequalities for stationary Navier-Stokes equations but without making use of the theory of pseudo-monotone operators. To do so, we consider a classical assumption, due to J. Rauch, which constrains us to make a slight change on the definition of a solution. The Rauch assumption,, although insure the existence of a solution, does not allow the conclusion that the non-convex functional is locally Lipschitz. Moreover, two dependence results are proved, one with respect to changes of the boundary condition and the other with respect to the density of external forces. The later one will be used to prove the existence of an optimal control to the distributed parameter optimal control problem where the control is represented by the external forces.

show abstract

Frictional Contact Problems for Steady Flow of Incompressible Fluids in Orlicz Spaces

Cited by 2 publications

References 55 publications

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Hemivariational Inequality for Navier–Stokes Equations: Existence, Dependence, and Optimal Control

Contact Info

Product

Resources

About