Degenerate Parabolic Diffusion Equations: Theory and Applications in Climatology - Vlachos, Panagiotis (University of Waterloo, 2024)
Diffusion models describe the spread of particles, energy, or other entities within a medium. Perturbations of mechanical systems, random walks(discrete case), and Brownian motion(continuous-time stochastic process) are some classical methods used to model diffusion. Among these, those generated by stochastic processes have been extensively studied by employing the Fokker-Planck equation—a one-dimensional parabolic partial differential equation—to examine these systems by analyzing the probability density function. Given the incomplete theory surrounding degenerate diffusion equations, our objective is to generalize and expand existing results for degenerate diffusion processes by examining cases where weak degeneracy occurs at the boundaries, utilizing a Fokker-Planck-like equation. More precisely, we first address the well-posedness results, which ensure the existence and uniqueness of a solution and are critical for investigating other qualitative properties such as controllability, observability, stabilization, and optimal control. Additionally, we explore the interval of the existence or absence of stationary states, which is fundamental in the analysis of mechanical or physical systems. To this end, we examine sufficient conditions for both the non-existence and existence of stationary points. Furthermore, to verify and illustrate our analytical results, we delve into the Budyko-Sellers model, a climate model, providing results on its well-posedness and addressing the inverse problem of determining the insolation function. Throughout this study, we primarily employ semigroup theory, op- erator and functional analysis, and weighted Sobolev spaces to manage the non-ellipticity of the diffusion and well-posedness of the parabolic equation, while using the theory of Lyapunov functions to ensure the existence of stationary states.
Keywords: Fokker-Planck equation; degenerate diffusion; invariant measures; semigroup theory; Lipschitz stability; climate model; steady-state; parabolic partial differential equations