*Alexandra Vict. ANTONIOUK*

Department of Nonlinear Analysis,

Institute of Mathematics of NAS of Ukraine,

Tereshchenkivs'ka Str. 3,

01601 Kyiv-4, UKRAINE

E-mail: antoniouk@imath.kiev.ua

**Regularity properties of the infinite dimensional evolutions,
related with anharmonic lattice systems**

**Abstract:**

In this talk we discuss the C^{¥} properties of
evolution for the infinite system of interacting particles. Such evolutions are described
in the terms of semigroup, generated by second order differential operator of infinite
number of variables. For anharmonic systems the corresponding operator has unbounded
essentially non-Lipschitz coefficients. In this case the semigroup is not strongly
continuous in time in the spaces of continuously differentiable functions and standard
analytic techniques become inapplicable.

Basing on the observation about nonlinear symmetry of high order variational
calculus, we demonstrate the smooth and raise of smoothness properties of such infinite
dimensional evolutions in the spaces of continuously differentiable functions. The main
attention is devoted to the influence of nonlinearity of the initial system on the
topologies of regularity.

In fact we provide a detail description of regularity properties for the
latice approximations of :P(f)_{d}: Euclidean field
models.