On Expansion of Regularity of Nonlinear Evolution Equations by Means of Dilation Symmetry

The paper present a dilation symmetry based approach to expansion of regularity of nonlinear evolution equations. In particular, it is shown that a symmetry of an operator, which describes a right-hand side of a non-linear evolution equation, is inherited by solutions of this equation. In the case of dilation symmetry, the latter implies that global-in-time existence of solutions for small initial data always imply global-in-time existence of solutions for large initial data. As an example, we consider the problem of expansion of regularity of the Navier-Stokes equations (in $\R^n$) accepting that the existence of global-in-time solutions for small initial data is already proven