Weak Solutions and Optimal Control of Hemivariational Evolutionary Navier-Stokes Equations under Rauch Condition

Mahdioui, Hicham; Aadi, Sultana Ben; Akhlil, Khalid

doi:10.1155/2020/6573219

Cited by 4 publications

(1 citation statement)

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“…This theory was initiated by Panagiotopoulus as a generalization of the variational inequality theory [29][30][31]. Hemivariational inequalities are suitable to model physical and engineering problems where multivalued and nonmonotone constitutive laws are involved (see, e.g., [22,23] and the references therein). The main tool in these formulations is the generalized gradient of Clarke and Rockafellar [8,10,34].…”

Section: Introductionmentioning

confidence: 99%

Multivalued nonmonotone dynamic boundary condition

et al. 2021

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In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

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