This paper treats the stochastic Navier-Stokes equations (SNSE) for incompressible fluids: \[ \begin{aligned} d \tilde{u}_{\varepsilon} - \{ \nu \cdot \varDelta \tilde{u}_{\varepsilon} - ( \tilde{u}_{\varepsilon} \cdot \nabla ) \tilde{u}_{\varepsilon} - \nabla \tilde{p}_{\varepsilon} \}\, dt &= \tilde{f}_{\varepsilon}\, dt + \tilde{G}_{\varepsilon} d \tilde{W}_{ \varepsilon} (t) \quad &\text{in} \quad Q_{\varepsilon} \times (0,T), \\ div\, \tilde{u}_{\varepsilon} &= 0 \qquad \qquad &\text{in} \quad Q_{\varepsilon} \times (0,T), \end{aligned} \tag{1} \] in thin spherical shells \[Q_{\varepsilon} := \{ y \in {\mathbb R}^3 : 1 \leqslant \vert y \vert \leqslant 1 + \varepsilon \}, \quad \text{where} \quad 0 < \varepsilon < \frac{1}{2}, \] along with free boundary conditions \[ \begin{aligned} \tilde{u}_{\varepsilon} \cdot \mathbf{n} = 0, \quad curl\, \tilde{u}_{\varepsilon} \times \mathbf{n} &= 0 \qquad &&\text{on} \quad \partial Q_{\varepsilon} \times (0,T), \\ \tilde{u}_{\varepsilon} (0, \cdot) &= \tilde{u}_0^{\varepsilon} &&\text{in} \quad Q_{\varepsilon}, \end{aligned} \] where \(\tilde{u}_{\varepsilon} = ( \tilde{u}_{\varepsilon}^r, \tilde{u}_{\varepsilon}^{\lambda}, \tilde{u}_{\varepsilon}^{\varphi})\) is the fluid velocity field, \(p\) is the pressure, \(\nu > 0\) is a fixed kinematic viscosity, \(\tilde{u}_0^{\varepsilon}\) is a divergence free vector field on \(Q_{\varepsilon}\), and \(\mathbf{n}\) is the unit outer normal vector to the boundary \(\partial Q_{\varepsilon}\), and \(\tilde{W}_{\varepsilon}(t)\), \(t \geq 0\), is an \({\mathbb R}^N\)-valued Wiener process in some probability space \(( \Omega, {\mathcal F}, {\mathbb F}, {\mathbb P} )\). \(\tilde{G}_{\varepsilon}\) is a family of maps : \({\mathbb R}_+ \to \tau_2( {\mathbb R}^N; H_{\varepsilon} )\) with \[ H_{\varepsilon} := \{ u \in {\mathbb L}^2(Q_{\varepsilon} ) : div \, u =0 \text{ in } Q_{\varepsilon}, \;\; u \cdot \mathbf{n} = 0 \text{ on } \partial Q_{\varepsilon} \}, \] where \(\tau_2\) denotes the Hilbert-Schmidt class. Assume that \(\tilde{f}_{\varepsilon} \in L^p ( [0,T]; V_{\varepsilon}' )\) for \(\varepsilon \in (0, 1]\) such that for some \(C > 0\), \[ \int_0^T \Vert \tilde{f}_{\varepsilon} (s) \Vert_{ V_{\varepsilon}'}^p\, ds \leqslant C e^{p/2}. \] \(V_{\varepsilon} = {\mathbb W}^{1,2}( Q_{\varepsilon} ) \cap H_{\varepsilon}\) and \(V_{\varepsilon}'\) is a dual space of \(V_{\varepsilon}\). The main statement establishes the convergence of the radial averages of the martingale solution of the 3D stochastic equation (1), as the thickness of the shell \(\varepsilon \to 0\), to a martingale solution \(u\) of the following stochastic Navier-Stokes equations on the unit sphere \({\mathbb S}^2 \subset {\mathbb R}^3\): \[ \begin{aligned} d u - \{ \nu \varDelta' u - ( u \cdot \nabla') u - \nabla' p \}\, dt &= f dt + G d W \quad &&\text{in} \quad {\mathbb S}^2 \times (0,T), \\ div' \, u &= 0 \qquad &&\text{in} \quad {\mathbb S}^2 \times (0,T), \\ u(0, \cdot) &= u_0 &&\text{in} \quad {\mathbb S}^2, \end{aligned} \] where \(u = ( u_{\lambda}, u_{\varphi})\) and \(\varDelta'\) (resp. \(\nabla'\) ) is the Laplace-de Rham operator (resp. the surface gradient) on \({\mathbb S}^2\) respectively. A point \(\eta \in Q_{\varepsilon}\) could be represented by the Cartesian coordinates \(\eta = (x,y,z)\) or \(\eta = (r, \lambda, \varphi)\) in spherical coordinates where \[ x= r \sin \lambda \cos \varphi, \quad y= r \sin \lambda \sin \varphi, \quad z = r \cos \lambda \] for \(r \in (1, 1+ \varepsilon)\), \(\lambda \in [0, \pi]\), and \(\varphi \in [0, 2 \pi)\).
Here is the main result:
Theorem. Let \((\Omega, {\mathcal F}, {\mathbb F}, {\mathbb P}, \tilde{W}_{\varepsilon}, \tilde{u}_{\varepsilon} )\) be a martingale solution of (1). Then the averages in the radial direction of this martingale solution converge to a martingale solution \(( \hat{\Omega}, \hat{ {\mathcal F} }, \hat{ {\mathbb F} }, \hat{ {\mathbb P} }, \hat{W}, \hat{u} )\) in \(L^2 ( \hat{ \Omega} \times [0,T] \times {\mathbb S}^2 )\).
For other related works, see, e.g. [Z. Brzeźniak et al., J. Math. Anal. Appl. 426, No. 1, 505–545 (2015; Zbl 1322.60102)] for random dynamical systems generated by stochastic Navier-Stokes equations on a rotating sphere, [Z. Brzeźniak et al., J. Math. Fluid Mech. 20, No. 1, 227–253 (2018; Zbl 1390.35260)] for random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere, [Z. Brzeźniak et al., Ann. Probab. 45, No. 5, 3145–3201 (2017; Zbl 1388.60107)] for invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains.