Topics in Degenerate Evolution Equations and Applied Mathematics

Hele- Shaw细胞由两个水平的，稍微分开的，平行的板组成，形成一个二维带，充满粘性流体（比如油）。然后，通过将粘性较低的流体（如水）注入通道，将油除去。在最初形成几个侵入的手指后，渗透流体达到稳定状态并以单个手指的形式存在。在数学上，人们在油所占的集合内寻找调和函数，在水所占的集合上消失。在分离两种流体的自由边界上，其中一种施加了保证质量守恒的运动学条件。50年代末，Saffman和Taylor明确地计算了入侵手指的一系列轮廓，参数化为手指的渐近上游宽度。然而，实验数据表明，这样的宽度总是通道宽度的1/2。自然界选择与参数的1/2值相对应的解的数学和物理机制尚不清楚。我们已经证明，在所有的Saffman-Taylor显式解中，对应于该参数值的1/2的解使流体穿过侵入手指尖端的通道的推力最大化。这个非变分的问题，通过白池型变换，被重新塑造成一个具有变分形式的问题。问题的非变分性质，设置在无界区域，是由一个精确的描述在无穷远处的解的渐近行为的相应的非线性椭圆方程。这种估计是通过哈纳克不等式的非标准应用和手指“鼻子”的识别来实现的。我们还表明，对于参数的1/2值，手指的运动是通过平均曲率发生的。我们将努力将这两个特性联系起来并加以理解。在另一个方向上，我们将研究Buckley—Leverett系统解的局部行为和唯一性。这是一个由两个退化抛物方程（在主部分）和奇异抛物方程（在低阶项）组成的系统。简并产生双曲-抛物线性质。Kruzkov观察到这种系统边值问题的唯一性与解的正则性有关。我们打算利用与双曲抛物问题有关的新思想和我们长期以来对退化进化方程解的正则性的研究，来研究这种解的唯一性。从理论上讲，我们将继续研究退化抛物方程解的Harnack型估计和变分学中的拟极小值。Hele-Shaw问题模拟的是石油渗入水中的过程。它的重要性源于它在开采被困在层状岩石土壤中的石油方面的应用（因此采用了二维模型）。物理上，我们观察到穿透手指的渐近宽度是通道宽度的1/2。我们试图理解的自然问题是，为什么大自然选择了这样一个值，以及这种特定选择发生的潜在数学和物理原因是什么。在同样的物理应用领域，人们会问两种流体是否以一种独特的方式渗透到另一种流体中（巴克利-莱弗里特系统），如果不是，那么可能缺乏独特性的原因是什么。支持这些物理问题的数学涉及对退化和/或奇异演化偏微分方程解的局部行为的精细估计，例如哈纳克不等式。

A Hele--Shaw cell consists of two horizontal, slightly separated, parallel plates, forming a 2-dimensional strip and filled with a viscous fluid (say for example oil). The oil is then removed by forcing a less viscous fluid (say water) into the channel. After an initial formation of several invading fingers, the penetrating fluid reaches a steady state and takes the form of a single finger. Mathematically one seeks a harmonic function within the set occupied by oil and vanishing on the set occupied by water. On the free boundary separating the two fluids, one imposed a kinematic condition guaranteeing conservation of mass. Saffman and Taylor in the late '50's computed explicitly a family of profiles of the invading finger, parameterized by the asymptotic upstream width of the finger. Experimental data however show that such a width is always 1/2 of the width of the channel.The mathematical and physical mechanism by which Nature selects the solution corresponding to the value 1/2 of the parameter, is not well understood. We have shown that among all the Saffman-Taylor explicit solutions, the one corresponding to the value 1/2 of the parameter, maximizes the thrust of the fluid across the channel at the tip of the invading finger. The problem, which is non--variational, is recast into one that has a variational form, through a Baiocchi-type transformation. The non-variational nature of the problem, set in an unbounded domain, is accounted for by a precise description of the asymptotic behavior at infinity of the solutions of the corresponding non-linear elliptic equation. Such estimation is achieved through non-standard applications of the Harnack Inequality and the identification of the 'nose' of the finger. We have also shown that for the value 1/2 of the parameter the motion of the finger occurs by mean curvature. An effort will be made to connect and understand these two features. In another direction, we will investigate local behavior and uniqueness of solutions to the Buckley--Leverett system. This is a system of two degenerate (in the principal part) and singular (in the lower order terms) of parabolic equations. The degeneracy yields a hyperbolic-parabolic behavior. Kruzkov observed that uniqueness of boundary value problems for such a system is linked to the regularity of the solutions. We intend to use recent ideas developed in connection with hyperbolic--parabolic problems and our long standing investigations on the regularity of solutions for degenerate evolution equations, to investigate the uniqueness of such solutions. More theoretically, we will continue our investigations on Harnack--type estimates for solutions of degenerate parabolic equations and quasi--minima in the Calculus of Variations. The Hele-Shaw problem simulates the penetration of oil into water. It's importance stems from its applications to the recovery of oil trapped into layered rocky soil (hence the 2-dimensional model). Physically one observes that the asymptotic width of the penetrating finger is 1/2 of the width of the channel. The natural questions we are attempting to understand is why Nature selects such a value and what's the underlying mathematical and physical reason for such a specific selection to occur. On the same realm of physical application one asks whether two fluids one penetrating into another (Buckley-Leverett system) do so in a unique manner, and if not, what is the reason the a possible lack of uniqueness. The supporting mathematics to such physical issues involves fine estimates of the local behavior of solutions of degenerate and/or singular evolution partial differential equations, such as for example the Harnack inequality.