Analytical Aspects of Some Non-Linear Mathematical Models

9706260卡法雷利该项目涉及自由边界问题和最优运输。当本构关系或守恒量(温度、压力、密度)在所考虑的变量的某个值上不连续地改变其行为时，自由边界问题自然产生。典型的例子有固-液界面相，火焰传播中的燃烧-未燃烧混合物，以及在多孔介质中的流动。了解溶液及其界面的几何形状和稳定性对于选择和评估模拟方法以及理解模型本身都很重要。最优输运理论讨论了如何以最便宜的方式将变量x从S状态转移到状态T(或者自然如何做到这一点)。变量x可以表示货物、受训人员、信息，或者，在准地转模型中，表示气团的最佳配置。这些模型与非线性偏微分方程间的联系还需要进一步探讨。重点讨论了两个领域的非线性分析的几个基本问题：快速和突变、分配和最优运输理论。许多领域都会发生快速而突然的转变。典型的例子有熔化或凝固过程中固液界面的演化，气体或流体在多孔介质中的传播，或者连续介质力学中某些材料的弹性到塑性转变。在金融数学领域，有一个变量的阈值状态，在该阈值状态之后，期权被执行，商品被重新进货，或者简而言之，当做出“不连续”的决定时。我们研究了这类过程的稳定性、数值模拟的方法等。第二个感兴趣的领域是建立最优运输模型的方程。例如，当货物必须从一组地点(仓库)运送到另一组地点(分发点)时，就会出现这种情况。当然，我们希望将运输成本降至最低。不足为奇的是，控制这个问题的基本方程在气象学(锋面形成的准地转方程)和不同流体流动(润滑、流体中的气泡等)的数值模拟中重现，在气象学中，气团以最佳的方式“传输”自己。我们希望这种方法将提供对这一现象的新理解和更好的计算技术。

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.