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.