In this paper, we introduce the g-Navier–Stokes equations with time-fractional derivative of order
α
∈
(
0
,
1
)
{\alpha\in(0,1)}
in domains of
ℝ
2
{\mathbb{R}^{2}}
. We then study the existence and uniqueness of weak solutions by means of the Galerkin approximation. Finally, an optimal control problem is considered and solved.
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.
The Bénard problem consists in a system that couples the well-known Navier–Stokes equations and an advection-diffusion equation. In thin varying domains this leads to the g-Bénard problem, which turns out to be the classical Bénard problem when g is constant. The main goal of this paper is to, first of all, introduce the g-Bénard problem with time-fractional derivative of order $\alpha \in (0,1)$
α
∈
(
0
,
1
)
. This formulation is new even in the classical Bénard problem, that is with constant g. The second goal of this paper is to prove the existence and uniqueness of a weak solution by means of the Faedo–Galerkin approximation method. Some recent works on time-fractional Navier–Stokes equations have opened new perspectives in studying variational aspects in problems involving time-fractional derivatives.
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