arXiv:1811.07885v1 [math.AP] 17 Nov 2018 STOCHASTIC NAVIER-STOKES EQUATION ON A 2D ROTATING SPHERE WITH STABLE LÉVY NOISE: WELL-POSEDNESS AND INVARIANT MEASURES LEANNE DONG ABSTRACT. In this paper we first prove the existence and uniqueness of weak and strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by stable Lévy noise. Then we show the existence of invariant measures under the assumption of finite dimensional noise. 1. INTRODUCTION The deterministic Navier-Stokes system on the rotating sphere serves as a fundamental model that arises naturally in large scale atmospheric dynamics. Many authors have studied the NSE on the unit spheres. Notably, Il’in and Filatov [22, 20] considered the existence and uniqueness of solutions to these equations and the estimation of the Hausdorff dimension of their global attractors [21]. Teman and Wang studied the inertial forms of NSEs on the sphere while Teman and Ziane proved that the NSE on a 2D sphere is a limit of NSE defined a spherical cell [33]. In another direction, Cao, Rammaha and Titi proved the Gevrey regularity of the solution and found an upper bound on the asymptotic degrees of freedom for the long-time dynamics [10]. This paper is concerned with the following stochastic Navier-Stokes equations (SNSE) on a 2D rotating sphere: ∂ t u + ∇ u u−νLu + ω×u + ∇p = f + η(x,t ), div u =0,u(0)= u 0 (1.1) where L is the stress tensor, ω is the Coriolis acceleration, f is the external force and η is the noise process that can be informally described as the derivative of an H -valued Lévy process. Rigorous definitions of all relevant quantities in this equation will be given in section 2 and 3. The question of well-posedness for equation (1.1) with additive Gaussian noise has been studied in [2]. The new features in this paper are the following. First, we prove that given L 4 -valued noise, V ′ -valued forcing f and small H -valued initial data, there exists an uniqueness global weak (variational) solution which depends continuously on initial data. Moreover, with increased regularity of forcing and initial data, we prove an unique strong (PDE) solution for the abstract stochastic Navier-Stokes equations on the 2D unit sphere perturbed by stable Lévy noise. The existence time interval depends on the regularity of force and the assumption of the noise. Finally, deduce the existence of invariant measure for the SNSE and establish measure support. The paper is organised as follows. In section 2, we review the fundamental mathematical theory for the deterministic Navier-Stokes equations (NSE) on the sphere. We state some keep results without proofs. In section 3, we define the SNSE on the spheres. We start with some analytic facts; we introduce the driving noise process, which is a stable Lévy noise via subordination. The SNSE is then decomposed into an Ornstein Uhlenbeck (OU) process (associated with the linear part of the SNSE) and a shift-invariant subset of full measure is identified that satisfies the Marcinkiewicz strong law of large number. In section 4, we prove there exists global weak solution using the usual Galerkin approximation based on vector spherical harmonic series expansion. (see the proof of Theorem 3.2.5) Moreover, uniqueness is proven using the classical argument in the Date: School of Mathematics and Statistics, The University of Sydney. November 21, 2018. 1