Topics in Degenerate and Singular Parabolic Equations and Homogenization

简并和奇异抛物型方程以及齐次化主题

• 批准号: 1265548
• 负责人: Emmanuele DiBenedetto
• 金额: $ 19.09万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265548&HistoricalAwards=false
• 关键词: Topics; Degenerate; Singular; Parabolic; Equations

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.

扩散过程是由它们的扩散率来调节的，扩散率是扩散发生速度的“标尺”。这些现象包括传热、气体膨胀、流体流动、牛顿或非牛顿、均匀或复合介质、布朗运动、细胞内分子的运动等。在所有这些过程中，扩散率随扩散量本身（温度、气体或流体密度等）和/或其空间梯度的函数而变化。如果扩散系数在某一点变为零，扩散停止，现象在该点发生简并。如果它变成无界，扩散是无限快的，在那一点现象是奇异的。模拟这些现象的演化偏微分方程（PDEs）很少能被明确地求解，并且表现出数学行为，这是人们不太了解的。该项目旨在探索这类偏微分方程解的局部和全局行为。利用测量理论技术和harnack型不等式研究了连续性、可微性、奇点的突然消失和生成问题。其核心思想是，这些扩散过程以其自身固有的抛物线几何形状演变，并结合了其扩散的不断演变的“规范”。偏微分方程中的均质化是理解具有周期性结构的复合材料（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个尺度的周期性结构域中。第二信使钙和环鸟苷单磷酸（cGMP）在脊椎动物视网膜的视杆细胞和视锥细胞中的扩散就是一个例子。视杆细胞和视锥细胞都有很厚的层状结构，其中的层是折叠的膜。将计算第二信使在锥中的扩散的均匀化极限。分析的难点在于，当层/褶皱的厚度趋于零时，该域就会断开。均质化的极限允许计算点，第二信使钙和cGMP的浓度。反过来，它们又在褶皱的边界局部控制光子进入锥体所产生的电流。虽然这项研究本质上是理论性的，但所考虑的偏微分方程源于物理模型，如非混相流体（水-油）、非牛顿流体（浓流体）、相变（水-冰-气）、传热（热屏蔽的燃烧）和数学生物学中的一些问题（活细胞表面分子的运动）。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。粒子通过扩散在活细胞表面移动，以实现特定的功能。例如，受体捕获细胞外部的信号并将其传递到细胞内部，从而开始生化过程。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转导。在锥体的均质化过程中，第二信使钙和cGMP是外部信号（光子）的换能器，以调节皱褶表面离子通道产生的电流。这些扩散过程的改变会导致视觉信号级联功能不正常，从而导致病变（例如年龄相关性黄斑变性）。视锥细胞非常脆弱，很难进行实验，因此需要对它们的行为进行数学建模。