Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations.

We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent . We first prove the existence of possibly infinitely many bifurcations from the spatially homogeneous steady state. We then focus our attention on the associated free energy, proving existence of minimisers and even uniqueness for sufficiently weak interactions.

Instability, Rupture and Fluctuations in Thin Liquid Films: Theory and Computations.

Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets.