Topics in Degenerate and/or Singular Evolution and Applied Mathematics

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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.