Topics in Degenerate and/or Singular Evolution and Applied Mathematics

本研究的主要问题是关于某些退化和/或奇异演化方程的解的局部和全局结构，包括那些（Buckley—Leverett系统）具有临界（Hadamard）增长条件的低阶项。特别的问题包括解的局部正则性，固有的哈纳克估计，边界行为，以及变分中的拟极小值。对于具有对数奇点的演化方程，将研究柯西问题的可解性。在高于2维的空间中，这取决于对可能解所允许的渐近行为的初步描述，以及相应椭圆方程的可解性。这是一个严重不适定的问题，因为，例如，紧支持的数据不能生成解决方案。可解性取决于在一个单点近似解产生一致下界的可能性。那么我们开发的全球哈纳克估计将产生整个空间的下界。这个过程的主要困难是没有一个自然的拓扑来近似数据。两种不混相流体在多孔介质中的流动（Buckley—Leverett模型）是用带有奇异项的演化方程来模拟的。有人问，尽管内能跳跃，饱和是否保持连续。这就等于说，两种流体的接触界面是以这样一种方式发生的，即质量是局部守恒的。一类奇异方程是在平面上扩散的胶体薄膜的缓慢演化过程中产生的。奇点是对数的，是由范德华力引起的。随着膜的扩散，它可能会自发破裂。由于薄膜被用作保护金属表面的涂层，人们对这一过程的演变以及可能对破裂现象的某种理解感兴趣。这些方程的可解性，它们的结构，以及对断裂现象的可能理解将被研究。

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.