Non-linear diffusion in RD and in Hilbert Spaces, a Cylindrical/Functional Integral Study

TL;DR

This paper proves the existence and uniqueness of solutions for non-linear diffusion equations in finite and infinite-dimensional spaces, employing functional integral methods relevant to physics and complex systems.

Contribution

It introduces a rigorous approach to solving non-linear diffusion equations using functional and cylindrical integral techniques, extending to infinite-dimensional Hilbert spaces.

Findings

Existence and uniqueness of weak solutions for finite-dimensional non-linear diffusion models.

Development of path-integral solutions for these diffusion equations.

Functional integral solutions for linear diffusion in infinite-dimensional Hilbert spaces.

Abstract

We present a proof for the existence and uniqueness of weak solutions for a cut-off and non cut-off model of non-linear diffusion equation in finite-dimensional space RD useful for modelling flows on porous medium with saturation, turbulent advection, etc. - and subject to deterministic or stochastic (white noise) stirrings. In order to achieve such goal, we use the powerful results of compacity on functional Lp spaces (the Aubin-Lion Theorem). We use such results to write a path-integral solution for this problem. Additionally, we present the rigourous functional integral solutions for the Linear Diffussion equation defined in Infinite-Dimensional Spaces (Separable Hilbert Spaces). These further results are presented in order to be useful to understand Polymer cylindrical surfaces probability distributions and functionals on String theory.