Stochastic Navier-Stokes equations on 2D rotating spheres with stable Lévy noise

Abstract

The aim of this dissertation is to study stochastic Navier-Stokes equations (SNSE) on 2D rotating spheres in Hilbert space perturbed by pure jump Lévy noise of β-stable type. The first goal is to establish the well-posedness of solutions to this class of equations. The second goal is to investigate qualitative questions on ergodicity, asymptotic behaviour and random dynamics. In Chapter 2, we review the analytic and probabilistic preliminary required to present the main results of the thesis. Then we introduce the background material on Hilbert space valued cylindrical Lévy noise via subordination of β-stable type. In Chapter 3, we prove the existence and uniqueness of solutions to the SNSE under suitable assumptions of noise and forcing and, in the second part, we deduce the existence of an invariant measure with measure support. Chapter 4 is devoted to the study of random dynamical systems generated by our SNSE. In particular, we prove that, with sufficient regularity, there exists a finite-dimensional random attractor for our SNSE. Moreover, such a random attractor supports a Feller Markov Invariant measure.