Young equations with singularities

TL;DR

This paper establishes existence and uniqueness of solutions to Young equations with singular initial data, using a novel sewing map approach to handle less regular initial conditions and extend the integral construction.

Contribution

It introduces a new sewing map for a class of increments, enabling the construction of Young convolution integrals with less regular initial data than previously possible.

Findings

Proves existence and uniqueness of solutions to Young equations with singular initial data.

Develops a new sewing map for a class of increments.

Extends the integral construction to cases with blow-up in the $X_\alpha$-norm.

Abstract

In this paper we prove existence and uniqueness of a mild solution to the Young equation $dy(t)=Ay(t)dt+\sigma(y(t))dx(t)$, $t\in[0,T]$, $y(0)=\psi$. Here, $A$ is an unbounded operator which generates a semigroup of bounded linear operators $(S(t))_{t\geq 0}$ on a Banach space $X$, $x$ is a real-valued $\eta$-H\"older continuous. Our aim is to reduce, in comparison to [4] and [1] (see also [2,5]) in the bibliography, the regularity requirement on the initial datum $\psi$ eventually dropping it. The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval $[a,b]\subset \mathbb R$ when the $X_\alpha$-norm of the function under the integral sign blows up approaching $a$ and $X_{\alpha}$ is an intermediate space between $X$ and $D(A)$.