Modelling, regularisation, and analysis of Dean–Kawasaki type equations

Abstract

We investigate modelling and analysis for a specific class of stochastic equations arising in fluctuating hydrodynamics: this class, which we refer to as the Dean Kawasaki (DK) class, is broadly concerned with the description of mesoscopic fluctuations in finite-size particle systems.

We focus on two notable members of this class. The first one, to which most of the thesis is devoted, is the DK equation. We revisit its original derivation from physics in a mathematically rigorous way, by considering particles of nite rather than atomic size. We do this in the two relevant cases of independent particles and of particles weakly interacting via a pairwise potential. In both cases, we derive a regularised DK model in the form of a stochastically perturbed wave equation. For this model we establish high-probability existence and uniqueness results by using small-noise techniques.

The issue of almost-sure positivity of solutions (a critical feature for the DK class) motivates the final part of the thesis: there, we study a second member of the class, namely, a stochastic thin- lm equation. We provide sufficient conditions on the interplay of stochastic noise and the source potentials in order to extend a positive local solution (defined up to a stopping time) up to any deterministic time, and we draw relevant analogies with the existing literature and with the DK equation.

Finally, we detail possible directions for future work.

Date of Award	30 Oct 2019
Original language	English
Awarding Institution	University of Bath
Supervisor	Tony Shardlow (Supervisor) & Johannes Zimmer (Supervisor)