Understand
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 - \sigma(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 H\"older 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 $\lambda_1 > \frac13$ is admissible.
- On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assuming that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$.