Regularity results for non-linear Young equations and applications

TL;DR

This paper establishes conditions for the existence and regularity of solutions to certain non-linear Young equations involving unbounded operators, independent of initial data smoothness, and explores their properties and invariance conditions.

Contribution

It provides new regularity results for non-linear Young equations with unbounded operators, including solution existence, blow-up rates, and invariance criteria, regardless of initial data smoothness.

Findings

Unique mild solutions are classical under certain conditions.

The blow-up rate of solutions near zero is characterized.

An integral representation and chain rule for solutions are derived.

Abstract

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.