Nonlinear elliptic and parabolic equations with nonlocal effects

具有非局部效应的非线性椭圆和抛物线方程

• 批准号: 1001629
• 负责人: Luis Silvestre
• 金额: $ 16.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001629&HistoricalAwards=false
• 关键词: Nonlinear; elliptic; parabolic; equations; nonlocal

This project concerns the analysis of elliptic and parabolic partial differential equations. The focus is on the regularity of solutions of equations whose diffusion is nonlocal, such as equations involving the fractional Laplacian. These equations arise, in particular, from models involving discontinuous stochastic processes. One important problem is to understand the equations arising from stochastic control problems and stochastic games involving discontinuous Levy processes. These equations are the fractional-order version of fully nonlinear elliptic and parabolic equations. The analysis of the regularity of integro-differential equations is not only interesting because it extends the theory of regularity of second-order partial differential equations, but also because it gives us a extra insight into what causes the regularization in elliptic and parabolic problems. Another mathematically interesting feature is that, since fractional diffusion has scaling properties different from those for second-order diffusion, the interplay between the first-order terms of the equation and the diffusion can become nontrivial at small scales. The principal investigator will study equations with advection and fractional diffusion, with an emphasis on the critical and supercritical regimes. Examples of this type of equations are the critical and supercritical quasi-geostrophic equation and the Burgers equation with fractional diffusion.The nonlocal equations presented in this project arise from models in engineering, finance and physics that involve long-range interactions. The most common situation in which this happens occurs in models involving discontinuous stochastic processes. For example, the price of a stock can suddenly jump from a value to a very different one. It has been suggested that it may be better to model stock prices with so-called discontinuous Levy processes than with diffusions. In that case, the equation involved in computing the value of an American option would be an obstacle problem for an integro-differential equation. Other nonlinear integro-differential equations arise from stochastic games, or from the physical phenomena known as anomalous diffusion. Nonlocal equations are also obtained from deterministic models. For example if one assumes that temperature is diffused quickly through the atmosphere, then this phenomenon creates a nonlocal diffusion effect on the surface of the earth.

这个项目涉及椭圆和抛物型偏微分方程的分析。重点是扩散是非局部的方程的解的正则性，例如涉及分数拉普拉斯算子的方程。这些方程产生，特别是，从涉及不连续随机过程的模型。其中一个重要的问题是理解随机控制问题和随机博弈中的不连续Levy过程所产生的方程。这些方程是完全非线性椭圆和抛物方程的分数阶版本。积分微分方程的正则性分析不仅是有趣的，因为它扩展了二阶偏微分方程的正则性理论，而且还因为它给了我们一个额外的洞察力，是什么导致了椭圆和抛物问题的正则化。另一个数学上有趣的特征是，由于分数阶扩散具有与二阶扩散不同的标度特性，因此方程的一阶项与扩散之间的相互作用在小尺度下可能变得非平凡。主要研究者将研究对流和分数扩散方程，重点是临界和超临界状态。这类方程的例子是临界和超临界准地转方程和分数扩散的Burgers方程。在这个项目中提出的非局部方程来自于工程，金融和物理中涉及长程相互作用的模型。这种情况最常见的情况发生在涉及不连续随机过程的模型中。例如，股票的价格可以突然从一个值跳到一个非常不同的值。有人建议，它可能是更好的模型股票价格与所谓的不连续利维过程比扩散。在这种情况下，计算美式期权价值的方程将成为积分微分方程的障碍问题。其他的非线性积分微分方程来自随机博弈，或者来自被称为反常扩散的物理现象。非局部方程也从确定性模型中获得。例如，如果假设温度在大气中迅速扩散，那么这种现象就会在地球表面产生非局部扩散效应。