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Abstract. An algorithm for continuous data assimilation for the two-dimensional Bénard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier-Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Bénard convection problem associated with the observed (finite dimensional projection of) velocity.MSC Subject Classifications: 35Q30, 93C20, 37C50, 76B75, 34D06.

Abstract. In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named determining form, in a space of trajectories which is isomorphic to C 1 b (R; R N ), for N large enough depending on the physical parameters of the Navier-Stokes equations. We present a unified approach based on interpolant operators that are induced by any of the determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, etc... There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, thus its solutions converge, as time goes to infinity, to the set of steady states of the determining form. The second is that these steady states of the determining form are identified, one-to-one, with the trajectories on the global attractor of the Navier-Stokes equations. It is worth adding that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.

We show that the way in which finite differences are applied to the nonlinear term in certain partial differential equations (PDES) can mean the difference between dissipation and blow up. For fixed parameter values and arbitrarily fine discretizations we construct solutions which blow up in finite time for two semi-discrete schemes. We also show the existence of spurious steady states whose unstable manifolds, in some cases, contain solutions which explode. This connection between the blow-up phenomenon and spurious steady states is also explored for Galerkin and nonlinear Galerkin semi-discrete approximations. Two fully discrete finite difference schemes derived from a third semidiscrete scheme, reported by Foias and Titi to be dissipative, are analysed. Both latter schemes are shown to have a stability condition which is independent of the initial data. A similar result is obtained for a fully discrete Galerkin scheme. While the results are stated for the Kuramoto-Sivashinsky equation, most naturally carry over to other dissipative PDES.