Topics in Degenerate and Singular Parabolic Equations and Homogenization

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Diffusion processes are regulated by their diffusivity, which is a "gauge" of how fast the diffusion takes place. These phenomena include heat transfer, expansion of gases, flow of fluids, Newtonian or not, in either homogeneous or composite media, Brownian motions, movement of molecules in cells, etc. In all of them the diffusivity changes as function of the diffusing quantity itself (temperature, gas or fluid density, etc.) and/or its spacial gradient. If the diffusion coefficient becomes zero at some point, the diffusion stops and the phenomenon is degenerate at that point. If it becomes unbounded, the diffusion is infinitely fast, and the phenomenon is singular at that point. The evolution Partial Differential Equations (PDEs) modeling these phenomena, seldom can be solved explicitly, and exhibit a mathematical behavior, which is not well understood. The project aims at exploring the local and global behavior of solutions of these classes of PDEs. Issues of continuity, differentiability, sudden vanishing and generation of singularities are investigated by means of measure theoretical techniques, and Harnack-type inequalities. The central idea is that these diffusion processes, evolve with their own intrinsic parabolic geometry, that incorporates the evolving "gauge" of its diffusion. Homogenization in PDEs is an essential tool in understanding the local behavior of composite materials with periodic structures, such as alloys. Several diffusion phenomena in biology occur in periodically structured domains exhibiting multiple scales. An example is the diffusion of the second messengers Calcium and cyclic Guanosine Mono Phosphate (cGMP), in rods and cones in the retina of vertebrates. Both rods and cones exhibit a thickly layered structure, where the layers are folded membranes. Homogenized limits will be computed for the diffusion of the second messengers in cones. The analytical difficulty is that the domain becomes disconnected as the thickness of the layers/folds goes to zero. The homogenized limit permits one to compute pointwise, the concentration of the second messengers Calcium and cGMP. These in turn control, locally, at the boundary of the folds, the current generated by photons of light entering a cone. While this investigation is theoretical in nature, the Partial Differential Equationss considered, originate from physical models such as immiscible fluids (water-oil), non Newtonian flows (thick fluids), phase transition (water-ice-gas), heat transfer (burning of thermal shields), and some issue in mathematical biology (motion of molecules on the surface of living cells). As such they are interdisciplinary, involving mathematicians, physicists engineers, and biologists. These investigations will contribute to a deeper understanding of the underlying physical and/or biological phenomena. The project aims also at introducing new theoretical/mathematical tools because of the non standard nature of these degenerate/singular diffusion phenomena. Particles move, by diffusion, on the surface of living cells to effect specific functions. For example a receptor, captures a signal from outside a cell and transmit it to its interior to begin a biochemical process. The space-time location and the speed of diffusion of a receptor affect the intensity of the signal and its transduction inside the cell. In the homogenization of cones, the second messengers Calcium and cGMP are the transducers of the outside signal (photons of light) to regulate the current generated by ionic channels on the surface of the folds. Alteration of these diffusion processes, causes improper functioning of the visual signaling cascade, leading to pathologies (for example age-related macula degeneration). Cones are very fragile and hard to experiment with, calling for mathematical modeling of their behavior.