The Porous Medium Equation: Multiscale Integrability in Large Deviations

Abstract

We consider a zero-range process $\hat{I}{\cdot }_t^N(x)$ with superlinear local jump rate, which in a hydrodynamic-small particle rescaling converges to the porous medium equation ${\partial }_{t}u=\frac{1}{2}\hat{I}”u^{\hat{I}}\pm ,\hat{I}\pm >1$. As a main result we obtain a large deviation principle in any scaling regime of vanishing particle size $\ddot{I}{‡}_N\to 0$. The key challenge is to develop uniform integrability estimate on the nonlinearity $(\hat{I}{\cdot }^N(x){)}^{\hat{I}}\pm$ in a situation where neither pathwise regularity nor Dirichlet-form based regularity is readily available. We resolve this by introducing a novel multiscale argument exploiting the appearance of pathwise regularity across scales.