Nonlinear elliptic and parabolic equations with nonlocal effects

这个项目涉及椭圆型和抛物型偏微分方程式的分析。重点研究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的方程。这些方程特别是从涉及不连续随机过程的模型中产生的。一个重要的问题是理解随机控制问题和涉及不连续Levy过程的随机对策所产生的方程。这些方程是完全非线性椭圆型和抛物型方程的分数阶形式。积分-微分方程正则性的分析不仅因为它扩展了二阶偏微分方程正则性的理论，而且还因为它使我们对椭圆和抛物问题的正则化有了更多的了解。另一个有趣的数学特征是，由于分数阶扩散具有不同于二阶扩散的标度性质，方程的一阶项和扩散项之间的相互作用在小尺度上可以变得不平凡。主要研究人员将研究具有平流和分数扩散的方程，重点是临界和超临界区域。这类方程的例子是临界和超临界准地转方程和具有分数扩散的Burgers方程。本项目中提出的非局域方程源于工程、金融和物理中涉及远程相互作用的模型。最常见的情况发生在包含不连续随机过程的模型中。例如，一只股票的价格可能会突然从一个价值跃升到一个非常不同的价值。有人建议，用所谓的不连续利维过程来模拟股票价格，可能比用扩散来模拟股票价格更好。在这种情况下，计算美式期权价值所涉及的方程将是积分-微分方程式的障碍问题。其他的非线性积分-微分方程式产生于随机博弈，或来自被称为反常扩散的物理现象。非局部方程也可以从确定性模型中得到。例如，如果一个人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。

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.