30 research outputs found

    Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications

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    As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and oceans, we study their time discretization by an implicit Euler scheme. From deterministic viewpoint the 3D Primitive Equations are studied with physically realistic boundary conditions. From probabilistic viewpoint we consider a wide class of nonlinear, state dependent, white noise forcings. The proof of convergence of the Euler scheme covers the equations for the oceans, atmosphere, coupled oceanic-atmospheric system and other geophysical equations. We obtain the existence of solutions weak in PDE and probabilistic sense, a result which is new by itself to the best of our knowledge

    Inviscid Limits for a Stochastically Forced Shell Model of Turbulent Flow

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    We establish the anomalous mean dissipation rate of energy in the inviscid limit for a stochastic shell model of turbulent fluid flow. The proof relies on viscosity independent bounds for stationary solutions and on establishing ergodic and mixing properties for the viscous model. The shell model is subject to a degenerate stochastic forcing in the sense that noise acts directly only through one wavenumber. We show that it is hypo-elliptic (in the sense of Hormander) and use this property to prove a gradient bound on the Markov semigroup

    Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

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    In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space XX which is acted on by any continuous semigroup {S(t)}tβ‰₯0\{S(t)\}_{t \geq 0}. Suppose that Β§(t)}tβ‰₯0\S(t)\}_{t \geq 0} possesses a global attractor A\mathcal{A}. We show that, for any generalized Banach limit LIMTβ†’βˆž\underset{T \rightarrow \infty}{\rm{LIM}} and any distribution of initial conditions m0\mathfrak{m}_0, that there exists an invariant probability measure m\mathfrak{m}, whose support is contained in A\mathcal{A}, such that ∫XΟ•(x)dm(x)=LIMTβ†’βˆž1T∫0T∫XΟ•(S(t)x)dm0(x)dt, \int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t, for all observables Ο•\phi living in a suitable function space of continuous mappings on XX. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when {S(t)}tβ‰₯0\{S(t)\}_{t \geq 0} does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space XX to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
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