Quasilinear rough partial differential equations with transport noise

We investigate the Cauchy problem for a quasilinear equation with transport rough input of the form $$\mathrm{d}$ u-\partial_i(a^{ij}(u)\partial_j u)$\mathrm{d}$ t =$\mathrm{d}$ $\mathbf{X}$_t^i(x)\partial_i u_t,$ $u_0\in L^2$ on the torus ${\mathbb{T}}^d$, where $\mathbf{X}$ is two-step enhancement of a family of coefficients $(X_t^i(x){)}_{i=1,\dots d}$, akin to a geometric rough path with H"older regularity $\alpha >1\mathrm{/}3.$ Using energy estimates, we provide sufficient conditions that guarantee existence in any dimension, and uniqueness in the case when $X$ is divergence-free. We then focus on the one-dimensional scenario, with slightly more regular coefficients. Improving the a priori estimates of the first results, we prove existence of a class of solutions whose spatial derivatives satisfy a Ladyzhenskaya-Prodi-Serrin type condition. Uniqueness is shown in the same class, by obtaining an $L^{\mathrm{\infty }}(L^1)$ estimate on the difference of two solutions. The latter is obtained by establishing a link with a certain backward dual equation combined with a (rough) iteration lemma `a la Moser.