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.

Since the discovery of quasicrystals by Schechtman et. al.
in 1984, quasi-periodic models in mathematical physics have formed an
active area of research. In particular, effects of quasi-periodicity
were investigated in a widely studied model of magnetism: the Ising
model (quantum and classical). Numerical and some analytic results
began to appear in the late '80s; however, most interesting
(numerical) results hitherto remained rigorously unconfirmed.Most of
the previous results relied on a connection with hyperbolic dynamical
systems.It is our aim to rigorously confirm previous numerical
observations, as well as to prove new results, by exploiting further
the aforementioned connection. In particular, we'll prove
multi-fractal structure of the energy spectrum of one-dimensional
quantum quasi-periodic Ising models. We'll also discuss its fractal
dimensions and measure.

This is the first in a series of two (or three) talks on partial (and normal) hyperbolicity. Partial hyperbolicity is in a sence a generalization of the notion of uniform hyperbolicity -- a well developed branch of smooth dynamical systems. In this talk we will begin with a motivation, definitions and some basic examples, laying the ground for the subsequent discussion of more advanced topics (mainly questions concerning generalization of resulrts of hyperbolic dynamics to partially hyperbolic systems).

I will discuss some recent results about Lp and Schauder estimates for a class of non-local elliptic equations. Compared to previous known results, the novelty of our results is that the kernels of the operators are not necessarily to be homogeneous, regular, or symmetric.