Analytical Aspects of Some Non-Linear Mathematical Models

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9706260 Caffarelli The project involves free boundary problems and optimal transportation. Free boundary problems arise naturally when a constitutive relation or a conserved quantity (a temperature, a pressure, a density) changes discontinuously its behavior across some value of the variables under consideration. Typical examples are solid-liquid interphases, burnt-unburnt mixtures in flame propagation, and flow in porous media. Understanding of the geometry and stability of the solution and its interphase is important to select and evaluate simulation methods, as well as to understand the models themselves. The theory of optimal transporation discusses how to transport a variable x from a state S to a state T in a least expensive fashion (or how nature would do so). The variable x could represent goods, trainees, information, or, as in the quasi-geostrophic model, optimal disposition of air masses. The links of these models to non-linear partial differential equations need much further exploration. The focus is on several fundamental problems of nonlinear analysis in two areas: fast and abrupt transitions, allocation and optimal transportation theory. Fast and abrupt transitions occur in many fields. Typical examples are the evolution of solid-liquid interphases in melting or solidification, the propagation of gases or fluids in porous media or the elastic to plastic transition in some materials in continuum mechanics. In the field of financial mathematics there is the threshold state of variables after which an option is executed, a good is restocked, or, in short, when a "discontinuous" decision is taken. We study the stability of such processes, the way to simulate them numerically, etc. The second area of interest consists of the equations that model optimal transportation. This arises for instance when goods must be delivered from one set of locations (warehouses) to another (distribution points). Of course we want to do it minimizing transp ortation costs. Not surprisingly, the basic equations governing this problem reappear in meteorology (the quasi-geostrophic equations of front formation), where air masses "transport" themselves optimally, and in the numerical simulation of different fluid flows (lubrication, bubbles in fluids, etc.). We hope that this approach will provide a new understanding of the phenomena and better computational techniques.