  We consider the following SPDE on a Gelfand-triple $(V, H, V^*)$: $$
  du(t)=A(t,u(t))dt+dI_t(u), \qquad u(0)=u_0\in H. $$
  Given certain local monotonicity, continuity, coercivity and growth
conditions of the operator $A:[0, T]\times V\to V^*$ and a sufficiently regular
operator $I$ we establish global existence of weak solutions.
  In analogy to the Young regime for SDEs, no probabilistic structure is
required in our analysis, which is based on a careful combination of monotone
operator theory and the recently developed Besov rough analysis by Friz and
Seeger. Due to the abstract nature of our approach, it applies to various
examples of monotone and locally monotone operators $A$, such as the
$p$-Laplace operator, the porous medium operator, and an operator that arises
in the context of shear-thickening fluids; and operators $I$, including
additive Young drivers $I_t(u) = Z_t-Z_0$, abstract Young integrals $I_t(u) =
\int_0^t \sigma(u_s)d X_s$, and translated integrals $I_t(u) = \int_0^t b(u_s -
w_s)ds$ that arise in the context of regularization by noise. In each of the
latter cases, we identify corresponding noise regimes (i.e. Young regimes) that
assure our abstract result to be applicable. In the case of additive drivers,
we identify the Brownian setting as borderline, i.e. noises which enjoy
slightly more temporal regularity are amenable to our completely pathwise
analysis.
