  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.
