Bulletin of the
Korean Mathematical Society BKMS

In this paper we study a nudging continuous data assimilation algorithm for the three-dimensional Leray-$\alpha$ model, where measurement errors are represented by stochastic noise. First, we show that the stochastic data assimilation equations are well-posed. Then we provide explicit conditions on the observation density (resolution) and the relaxation (nudging) parameter which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solution which is corresponding to these measurements, in terms of the variance of the noise in the measurements.

Keywords: Continuous data assimilation, stochastic Leray-$\alpha$ model, finite volume elements, interpolants, nudging

MSC numbers: Primary 35Q35, 60H15, 60H30, 93C20, 76B75, 37C50