Quasilinear rough partial differential equations with transport noise

Abstract

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 Hölder 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 à la Moser.

Published version; 42 pages