Quasilinear SPDEs via Rough Paths

Otto, Felix; Weber, Hendrik

Abstract

We are interested in (uniformly) parabolic PDEs with a nonlinear dependence of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: where P is the projection on mean-zero functions, and f is a distribution which is only controlled in the low regularity norm of for on the parabolic Hölder scale. The example we have in mind is a random forcing f and our assumptions allow, for example, for an f which is white in the time variable x2 and only mildly coloured in the space variable x1; any spatial covariance operator with is admissible. On the deterministic side we obtain a -estimate for u, assuming that we control products of the form and vf with v solving the constant-coefficient equation . As a consequence, we obtain existence, uniqueness and stability with respect to of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing f using stochastic arguments. For this we extend the treatment of the singular product via a space-time version of Gubinelli’s notion of controlled rough paths to the product , which has the same degree of singularity but is more nonlinear since the solution u appears in both factors. In fact, we develop a theory for the linear equation with rough but given coefficient fields a and and then apply a fixed point argument. The PDE ingredient mimics the (kernel-free) Safonov approach to ordinary Schauder theory.