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 $(X^i_t(x))_{i=1,\dots d}$, akin to a geometric rough path with H\"older regularity $\alpha>1/3.$ 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 $L^\infty(L^1)$ 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.