Regularity results for nonlinear Young equations and applications

Abstract

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