*Alexander Val. 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

**Nonlinear symmetries of variational calculus and regularity
properties of differential flows on** **non-compact manifolds**

**Abstract:**

This talk is devoted to the discussion of symmetries that arise in the
high order variational equations. Such symmetries lead to a new class of nonlinear
estimates on variations and permit the work in the case of essentially non-Lipschitz
nonlinear differential equations, i.e. when the classical Cauchy-Liouville- Picard
regularity scheme fails to work.

These estimates are applied to the problem of C^{¥}
differentiability with respect to the initial data and parameters for the nonlinear
differential flows on manifolds, that could also contain random terms. In particular, we
demonstrate that the geometrically correct study of regularity problems for nonlinear
flows on manifolds requires introduction of a new type variations with respect to the
initial data. These variations are defined via a natural generalization of covariant
Riemannian derivatives.

We also find how the curvature manifests in the structure of the high order
variational equations.