Topics in Degenerate and Singular Parabolic Equations and Homogenization

扩散过程由其扩散率调节，扩散率是扩散发生速度的“标尺”。这些现象包括热传递、气体膨胀、流体流动、牛顿或非牛顿流体、均质或复合介质、布朗运动、细胞中分子的运动等。在所有这些现象中，扩散率随扩散量本身（温度、气体或流体密度等）而变化。和/或其空间梯度。如果扩散系数在某一点变为零，则扩散停止，并且该现象在该点处退化。如果它变得无界，扩散是无限快的，并且在那一点上现象是奇异的。演化偏微分方程（PDE）模拟这些现象，很少可以显式求解，并表现出一个数学行为，这是不是很好地理解。该项目的目的是探索这些类偏微分方程的解决方案的本地和全球的行为。 利用测度理论和Harnack型不等式研究了连续性、可微性、突然消失和奇点的产生等问题。其中心思想是，这些扩散过程，演变与自己的内在抛物线几何，其中包括不断发展的“规范”的扩散。偏微分方程中的均匀化是理解具有周期结构的复合材料（如合金）局部行为的重要工具。生物学中的一些扩散现象发生在具有多尺度的周期性结构域中。一个例子是第二信使钙和环磷酸鸟苷（cGMP）在脊椎动物视网膜的视杆和视锥中的扩散。视杆细胞和视锥细胞都表现出厚层结构，其中层是折叠的膜。 将计算第二信使在视锥细胞中扩散的均匀化限度。分析的困难在于，当层/折叠的厚度变为零时，域变得不连续。均质化极限允许逐点计算第二信使钙和cGMP的浓度。这些反过来控制，局部地，在褶皱的边界，由光子进入圆锥体产生的电流。虽然这项研究是理论性的，但所考虑的偏微分方程源于物理模型，如不混溶流体（水-油），非牛顿流动（厚流体），相变（水-冰-气），传热（隔热屏的燃烧），以及数学生物学中的一些问题（活细胞表面上的分子运动）。因此，它们是跨学科的，涉及数学家，物理学家，工程师和生物学家。这些调查将有助于更深入地了解潜在的物理和/或生物现象。 该项目还旨在引入新的理论/数学工具，因为这些退化/奇异扩散现象的非标准性质。 颗粒通过扩散在活细胞表面移动以实现特定功能。例如受体，从细胞外部捕获信号并将其传递到细胞内部以开始生化过程。受体的时空位置和扩散速度影响信号的强度及其在细胞内的传递。在锥体的均质化中，第二信使钙和cGMP是外部信号（光子）的转换器，以调节褶皱表面上的离子通道产生的电流。 这些扩散过程的改变导致视觉信号级联的功能不正常，从而导致病理（例如年龄相关性黄斑变性）。圆锥体非常脆弱，很难进行实验，需要对其行为进行数学建模。

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.