Regularity results for non-linear Young equations and applications

In this paper we provide sufficient conditions which ensure that the non-linear equation $dy(t)=Ay(t)dt+\sigma(y(t))dx(t)$, $t\in(0,T]$, with $y(0)=\psi$ and $A$ being an unbounded operator, admits a unique mild solution which is classical, i.e., $y(t)\in D(A)$ for any $t\in (0,T]$, and we compute the blow-up rate of the norm of $y(t)$ as $t\rightarrow 0^+$. We stress that the regularity of $y$ is independent on the smoothness of the initial datum $\psi$, which in general does not belong to $D(A)$. As a consequence we get an integral representation of the mild solution $y$ which allows us to prove a chain rule formula for smooth functions of $y$ and necessary conditions for the invariance of hyperplanes with respect to the non-linear evolution equation.