Topics in Degenerate and/or Singular Evolution and Applied Mathematics

9706388 迪贝内代托 本次调查的主要问题涉及当地 的解的整体结构 退化和/或奇异演化方程，包括 Buckley-Leverett系统 阿达玛（Hadamard）生长条件。 特别问题包括 解的局部正则性，内在的 Harnack估计、边界行为和拟极小 在变分法中。对于演化方程 具有对数奇点的可解性 将研究Cauchy问题。在空间维度中 高于两个，这取决于初步的描述 可能解所允许的渐近行为， 以及相应椭圆方程的可解性 方程这是一个严重不适定的问题，因为， 例如，计算机支持的数据不会生成解决方案。 可解性取决于 一阶近似解的一致下界 单点。那么我们得出的全球Harnack估计 会在整个空间中得到一个下界的主要困难 在这个过程中， 拓扑结构来近似数据。 两种不混溶流体在多孔介质中的流动 （Buckley-Leverett模型）用演化方程建模 带有单数形式。有人问，尽管跳跃在 内部能量，饱和度保持连续。这将 相当于说两种流体的接触界面 以这样一种方式发生，即质量局部守恒。 在慢演化过程中出现了一类奇异方程 一层薄薄的胶状薄膜覆盖在平面上。的奇点 是对数的，并且是由于货车范德华力。 当薄膜扩散时，它可能会自发破裂。以来 该膜作为涂层应用以保护金属表面，一种是 对这一过程的演变感兴趣， 对断裂现象有了一些了解 这些方程的可解性，它们的结构， 将研究对断裂现象的理解。

9706388 DiBenedetto The main issues of this investigation concern the local and global structure of solutions of some classes of degenerate and/or singular evolution equations, including those (Buckley--Leverett system) bearing lower order terms with critical (Hadamard) growth conditions. Special issues include the local regularity of the solutions, intrinsic Harnack estimates, the boundary behavior, and Quasi--Minima in the Calculus of Variations. For evolution equations bearing logarithmic singularities the solvability of the Cauchy problem will be studied. In space-dimensions higher than two, this hinges upon a preliminary description of the asymptotic behavior permitted by the possible solutions, as well as the solvability of the corresponding elliptic equation. This is a severely ill-posed problem, as, for example, compactly supported data do not generate a solution. Solvability depends on the possibility of generating uniform lower bounds for approximating solutions at one single point. Then the global Harnack estimate we have developed would yield a lower bound in the whole space. The main difficulty in this process is that there is not a natural topology by which to approximate the data. The flow of two immiscible fluids in a porous medium (Buckley--Leverett model) is modeled by evolutions equations bearing singular terms. One asks whether, despite the jump in internal energy, the saturation remains continuous. This would amount to say that the interface of contact of the two fluids occurs in such a way that the mass is locally conserved. A class of singular equations arises in the slow evolution of thin colloidal films spread over a flat surface. The singularities are logarithmic and are due to the van der Waals forces. As the film spreads, it may undergo a spontaneous rupture. Since the film is applied as a coating to protect metal surfaces, one is interested in the evolution of such a process and possibly in some sort of understanding of the phenomenon of rupture. The solvability of these equations, their structure, and possibly an understanding of the rupture phenomenon will be investigated.