J. Math. Fluid Mech. 20 (2018), 227–253
2017 The Author(s).
This article is an open access publication
Journal of Mathematical
Random Attractors for the Stochastic Navier–Stokes Equations on the 2D Unit Sphere
Z. Brze´zniak, B. Goldys and Q. T. Le Gia
Abstract. In this paper we prove the existence of random attractors for the Navier–Stokes equations on 2 dimensional sphere
under random forcing irregular in space and time. We also deduce the existence of an invariant measure.
Mathematics Subject Classiﬁcation. Primary 35B41, Secondary 35Q35.
Keywords. Random attractors, Energy method, Asymptotically compact random dynamical systems, Stochastic Navier–
Stokes, Unit sphere.
Complex three dimensional ﬂows in the atmosphere and oceans are modelled assuming that the Earth’s
surface is an approximate sphere. Then it is natural to model the global atmospheric circulation on Earth
(and large planets) using the Navier–Stokes equations (NSE) on 2-dimensional sphere coupled to classical
thermodynamics . This approach is relevant for geophysical ﬂow modeling.
Many authors have studied the deterministic NSEs on the unit sphere. Notably, Il’in and Filatov [30,32]
considered the existence and uniqueness of solutions to these equations and estimated the Hausdorﬀ
dimension of their global attractors . Temam and Wang  considered the inertial forms of NSEs on
sphere while Temam and Ziane , see also , proved that the NSEs on a 2-dimensional sphere is a limit
of NSEs deﬁned on a spherical shell . In other directions, Cao et al.  proved the Gevrey regularity of
the solution and found an upper bound on the asymptotic degrees of freedom for the long-time dynamics.
Concerning the numerical simulation of the deterministic NSEs on sphere, Fengler and Freeden 
obtained some impressive numerical results using the spectral method, while the numerical analysis of a
pseudo- spectral method for these equations has been carried out in Ganesh et al. .
In our earlier paper  we analysed the Navier–Stokes equations on the 2-dimensional sphere with
Gaussian random forcing. We proved the existence and uniqueness of solutions and continuous dependence
on data in various topologies. We also studied qualitative properties of the stochastic NSEs on the unit
sphere in the context of random dynamical systems.
Building on those preliminary studies, in the current paper, we prove the existence of random attractors
for the stochastic NSEs on the 2-dimensional unit sphere. Let us recall here that, given a probability space,
a random attractor is a compact random set, invariant for the associated random dynamical system and
attracting every bounded random set in its basis of attraction (see Deﬁnition 4.4).
In the area of SPDEs the notions of random and pullback attractors were introduced by Brze´zniak et
al. , and by Crauel and Flandoli . These concepts have been later used to obtain crucial information
on the asymptotic behaviour of random , stochastic [2,16,17,26] and non-autonomous PDEs [13,33,36].
This work was partially supported by the Australian Research Council Project DP120101886.