Abstract
AbstractThe 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.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
Reference35 articles.
1. Aayadi, K., Akhlil, K., Ben Aadi, S., Mahdioui, H.: Weak solutions to the time-fractional g-Bénard equations. Preprint. arXiv:2011.01545v1
2. Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40, 6287–6303 (2007)
3. Alikhanov, A.A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46, 660–666 (2010)
4. Anh, C.T., Quyet, D.T.: g-Navier–Stokes equations with infinite delays. Vietnam J. Math. 40(1), 57–78 (2012)
5. Bae, H., Roh, J.: Existence of solutions of the g-Navier–Stokes equations. Taiwan. J. Math. 8(1), 85–102 (2004)