Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== Fix t∈(0,T]t\in(0,T]t∈(0,T]. Write ====

I(t)=∫0teν(t−s)ΔB(u(s),u(s)) ds.\mathcal I(t) = \int_0^t e^{\nu (t-s)\Delta} B(u(s),u(s))\,ds.I(t)=∫0teν(t−s)ΔB(u(s),u(s))ds.
Standard dominated-convergence arguments for Bochner integrals give continuity of t↦I(t)t\mapsto\mathcal I(t)t↦I(t) provided:
* the integrand is strongly measurable in sss, (true here because u(s)u(s)u(s) is continuous into XXX and the composition with bounded linear maps is continuous), and
* there is an integrable dominating function independent of small perturbations of ttt. The estimate above provides such a uniform integrable bound ∝(t−s)−α\propto(t-s)^{-\alpha}∝(t−s)−α when ttt ranges over a compact subinterval away from 000. One checks the contribution near the moving endpoint is small uniformly.

A standard calculation (write difference I(t+h)−I(t)\mathcal I(t+h)-\mathcal I(t)I(t+h)−I(t) and split the integral) shows I(⋅)\mathcal I(\cdot)I(⋅) is continuous on (0,T](0,T](0,T]; continuity at t=0t=0t=0 follows from small-time smoothing estimates and smallness of the integral on small intervals. Therefore I∈C([0,T];X)\mathcal I\in C([0,T];X)I∈C([0,T];X).