Quasilinear SPDEs via rough paths

Abstract

We are interested in (uniformly) parabolic PDEs with a nonlinear dependance 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: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - σ(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $C^{\alpha -2}$ for $\alpha >\frac{2}{3}$ 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 $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1+|{\partial }_1|{)}^{-{\lambda }_1}$ with ${\lambda }_1>\frac{1}{3}$ is admissible. On the deterministic side we obtain a $C^{\alpha }$-estimate for $u$, assuming that we control products of the form $v{\partial }_1^{2}v$ and $vf$ with $v$ solving the constant-coefficient equation ${\partial }_{2}v-a_0{\partial }_1^{2}v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f,vf,v{\partial }_1^{2}v)$ 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 $\sigma (u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u){\partial }_1^{2}u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.

arXiv comment65 pages