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We are interested in (uniformly) parabolic PDEs with a nonlinear dependanceof the leading-order coefficients, driven by a rough right hand side. Forsimplicity, we consider a space-time periodic setting with a single spatialvariable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0  \end{equation*} where $P$ is the projection on mean-zero functions, and $f$is a distribution and only controlled in the low regularity norm of $C^{\alpha-2}$ for $\alpha > \frac{2}{3}$ on the parabolic H\"older scale.  The example we have in mind is a random forcing $f$ and our assumptionsallow, for example, for an $f$ which is white in the time variable $x_2$ andonly mildly coloured in the space variable $x_1$; any spatial covarianceoperator $(1 + |\partial_1|)^{-\lambda_1 }$ with $\lambda_1 > \frac13$ isadmissible.  On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assumingthat we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solvingthe constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As aconsequence, we obtain existence, uniqueness and stability with respect to $(f,vf, v \partial_1^2v)$ of small space-time periodic solutions for small data. Wethen demonstrate how the required products can be bounded in the case of arandom forcing $f$ using stochastic arguments.  For this we extend the treatment of the singular product $\sigma(u)f$ via aspace-time version of Gubinelli's notion of controlled rough paths to theproduct $a(u)\partial_1^2u$, which has the same degree of singularity but ismore nonlinear since the solution $u$ appears in both factors. The PDEingredient mimics the (kernel-free) Krylov-Safanov approach to ordinarySchauder theory.

Quasilinear SPDEs via rough paths