Topics in Degenerate and/or Singular Evolution and Applied Mathematics

9706388本研究的主要问题涉及某些退化和/或奇异发展方程的解的局部和全局结构，包括那些含有临界(Hadamard)增长条件的低阶项的发展方程(Buckley-Leverett系统)。特殊问题包括解的局部正则性、固有Harnack估计、边界行为和变分中的拟极小问题。对于含有对数奇点的发展方程，我们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，以及相应的椭圆型方程的可解性。这是一个严重不适定的问题，因为，例如，紧密支持的数据不会生成解决方案。可解性取决于在一个点上产生逼近解的统一下界的可能性。然后，我们开发的全球哈纳克估计将在整个空间产生一个下限。这一过程中的主要困难是没有一个自然的拓扑结构来近似数据。两种不相容流体在多孔介质中的流动(Buckley-Leverett模型)由含有奇异项的演化方程来模拟。有人问，尽管内部能量跃升，但饱和度是否保持持续。这相当于说，两种流体的接触界面以这样一种方式发生，即质量是局部守恒的。在平面上铺展的胶体薄膜的缓慢演化过程中，出现了一类奇异方程。奇点是对数的，是由范德华力引起的。随着薄膜的扩散，它可能会发生自发破裂。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感兴趣，也可能对破裂现象有某种理解。我们将研究这些方程的可解性、它们的结构以及可能对破裂现象的理解。

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.