Regularity results for nonlinear Young equations and applications

Abstract

In this paper we provide sufficient conditions which ensure that the nonlinear equation mathrm{d}y(t)=Ay(t)mathrm{d}t+sigma (y(t))mathrm{d}x(t), tin (0,T], with y(0)=psi and A being an unbounded operator, admits a unique mild solution such that y(t)in D(A) for any tin (0,T], and we compute the blow-up rate of the norm of y(t) as trightarrow 0^+. We stress that the regularity of y is independent of 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.