Abstract
Incompressible Navier–Stokes equations on a thin spherical domain Q_varepsilon along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale solution of the stochastic Navier–Stokes equations on a sphere mathbb {S}^2 as the thickness converges to zero is established.
Highlights
For various motivations, partial differential equations in thin domains have been studied extensively in the last few decades; e.g. Babin and Vishik [4], Ciarlet [16], Ghidaglia and Temam [18], Marsden et al [37] and references there in
Raugel and Sell [44,45] proved the global existence of strong solutions to Navier–Stokes equations (NSE) on thin domains for large initial data and forcing terms, in the case of purely periodic and periodic-Dirichlet boundary conditions
Stochastic Navier–Stokes equations on thin spherical domains are introduced in Sect. 4 and a priori estimates for the radially averaged velocity are obtained which are later used to prove the convergence of the radial average of a martingale solution of stochastic NSE on thin spherical shell (see (1)–(5)) to a martingale solution of the stochastic NSE on the sphere (see (6)–(8)) with vanishing thickness
Summary
Partial differential equations in thin domains have been studied extensively in the last few decades; e.g. Babin and Vishik [4], Ciarlet [16], Ghidaglia and Temam [18], Marsden et al [37] and references there in. Qε, whose existence can be established as in the forthcoming paper [7] to the martingale solution of the stochastic Navier–Stokes equations on a two dimensional sphere S2 [9] as thickness ε of the spherical domain converges to zero. In this way we give another proof for the existence of a martingale solution for stochastic NSE on the unit sphere S2. Stochastic Navier–Stokes equations on thin spherical domains are introduced in Sect. 4 and a priori estimates for the radially averaged velocity are obtained which are later used to prove the convergence of the radial average of a martingale solution of stochastic NSE on thin spherical shell (see (1)–(5)) to a martingale solution of the stochastic NSE on the sphere (see (6)–(8)) with vanishing thickness
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