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.