Browsing by Author "Chueshov, Igor"

Now showing 1 - 3 of 3
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    Navier-Stokes, Fluid Dynamics, and Image and Video Inpainting
    (2001-06) Chueshov, Igor; Duan, Jinqiao; Schmalfuss, Bjorn
    Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional random dynamical systems. In these applications the convergence condition of the trajectories of an infinite dimensional random dynamical system with respect to a finite set of linear functionals is assumed to be either in mean or exponential with respect to the convergence almost surely. In contrast to these ideas we introduce a convergence concept which is based on the convergence in probability. By this ansatz we get rid of the assumption of exponential convergence. In addition, setting the random terms to zero we obtain usual deterministic results. We apply our results to the 2D Navier - Stokes equations forced by a white noise.
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    Probabilistic Dynamics of Two-Layer Geophysical Flows
    (2001-04) Chueshov, Igor; Duan, Jinqiao; Schmalfuss, Bjorn
    The two-layer quasigeostrophic flow model is an intermidiate system between the single-layer 2D barotropic flow model and the continuously stratified, 3D baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. The wind forcing acts only on the top layer. We consider the two-layer quasigeostrophic model under stochastic wind forcing. We first transformed this system into a coupled system of random partial differential equations and then show that the asymptotic probabilistic dynamics of this system depends only on the top fluid layer. Namely, in the probability sense and asymptotically, the dynamics of the two-layer quasigeostrophic fluid system is determinied by the top fluid layer, or, the bottom fluid layer is slaved by the top fluid layer. This conclusion is true provided that the Wiener process and the fluid parameters satisfy a certain condition. In particular, this latter condition is satisfied when the trace of the covariance operator of the Wiener process is controled by a certain upper bound, and the Ekman constant r is sufficiently large. Note that the generalized time derivative of the Wiener process models the fluctuating part of the wind stress forcing on the top fluid layer, and the Ekman constant r measures the rate for vorticity decay due to the friction in the bottom Ekman layer.
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    A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations
    (University of Minnesota. Institute for Mathematics and Its Applications, 2012-11) Chueshov, Igor