Topics in Degenerate Evolution Equations and Applied Mathematics

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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.