Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. In this lecture I will discuss the discrete data assimilation for the incompressible two-dimensional Navier-Stokes
equations. In both cases we obtain bounds on the time interval between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.

The Navier-Stokes system is the classical model for the motion of a viscous incompressible homogeneous fluid. Physically, the equations
express Newton's second law of motion and the conservation of mass. The unknowns are the velocity vector field u and the scalar pressure field p, while the volume forces f and the kinematic viscosity are given. In this talk, we address the spatial complexity and the local behavior of solutions to the three dimensional (NSE) with general non-analytic forcing. Motivated by a result of Kukavica and Robinson in [4], we consider a system of elliptic-parabolic type for a diference of two solutions (u1; p1) and (u2; p2) of (NSE) with the same Gevrey forcing f. By proving delicate Carleman estimates with the same singular weights for the Laplacian and the heat operator (cf. [1, 2, 3]), we establish a quantitive estimate of unique continuation leading to the strong unique continuation property for solutions of the coupled elliptic-parabolic system. Namely, we obtain that if the velocity vector fields u1 and u2 are not identically equal, then their diference
u1-u2 has finite order of vanishing at any point. Moreover, we establish a polynomial estimate on the rate of vanishing, provided the forcing f lies in the Gevrey class for certain restricted range of the exponents. In particular, the necessary condition for the result in [4] is satisfied; thus a finite-dimensional family
of smooth solutions can be distinguished by comparing a finite number of their point values.
This is a joint work with Igor Kukavica.

References
[1] M. Ignatova and I. Kukavica, Unique continuation and complexity of solutions to parabolic partial diferential equations with Gevrey coeficients, Advances in Diferential Equations 15 (2010), 953-975.
[2] M. Ignatova and I. Kukavica, Strong unique continuation for higher order elliptic equations with Gevrey
coeficients, Journal of Diferential Equations (submitted in August, 2010).
[3] M. Ignatova and I. Kukavica, Strong unique continuation for the Navier-Stokes equation with non-analytic forcing, Journal of Dynamics and Diferential Equations (submitted in January, 2011).
[4] I. Kukavica and J.C. Robinson, Distinguishing smooth functions by a finite number of point values, and a version of the Takens embedding theorem, Physica D 196 (2004), 45-66.

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture.

This is a joint work with Claude Bardos.