Viscosity solution methods in partial differential equations and applications

本文着重分析了一类非线性偏微分方程及其应用。将它们联系在一起的是粘度溶液的概念。该方案的一部分重点是与随机偏微分方程的最优控制相关的Hamilton-Jacobi-Bellman （HJB）型方程。与它们的控制相关的HJB方程是无限维空间中的方程。这类方程的理论还没有得到很好的发展。首席研究员（PI）在项目中研究它们，特别关注与特别感兴趣的问题相关的几个方程。其中一个问题是流体流动的最优控制，它可以被重新表述为确定性或随机Navier-Stokes方程的最优控制。另一个问题来自数学金融学，与期权定价有关。分析了“Black-Scholes”方程的无限维等价及其非线性版本，即“Black-Scholes- barenblatt”方程。Hilbert（或Banach）空间中的HJB方程是对偏微分方程驱动系统最优控制问题进行动态规划分析的关键。这些HJB方程必须从广义解的角度来研究。粘度解应该为这类方程提供正确的方法，所提出的研究应该是为无限维随机系统的最优控制奠定基础的重要组成部分。本文还研究了一类包含拟线性方程和某些几何型方程的推广的完全非线性非发散形式一致椭圆方程，这是椭圆理论中的一个重要类别。这类方程还没有被系统地研究过，特别是当它们在空间变量上是不连续的。这些方程没有经典解PI想把所谓的L^p粘度解的理论扩展到这类。特别是PI计划研究这类方程解的正则性问题。这是椭圆偏微分方程的一个主要开放问题，PI提出了几种可能的新方法来解决它，这些方法可能会产生新的有趣的技术。粘度解的概念是非线性偏微分方程的主要工具之一，它在最优控制、图像处理、移动前沿和相变、统计力学、经济学、数学金融等领域得到了广泛的应用。本文所描述的一些问题的研究动机来自于最优控制，特别是随机偏微分方程的控制。他们的理论正处于快速发展的状态，并受到来自物理学、人口生物学、化学、经济学和数学金融学的建模问题的推动。流体流动的最优控制问题是一个基本的工程问题，在燃烧理论、航空和水动力控制、托科马克磁聚变、海洋和大气预测等领域都有许多应用。与Musiela利率模型相关的问题来自于现代期权定价理论。该项目的研究应有助于发展偏微分方程的新方向，并对上述应用领域产生影响。

The proposal concentrates on the analysis of certain classes of nonlinear partial differential equations and their applications. Linking them together is the notion of viscosity solution. Part of the proposal focuses on equations of Hamilton-Jacobi-Bellman (HJB) type that are related to optimal control of stochastic partial differential equations. The HJB equations associated with their control are equations in infinite dimensional spaces. The theory of such equations is not well developed. The principal investigator (PI) studies them in the project paying special attention to several equations related to problems of particular interest. One of such problems is optimal control of fluid flow that can be reformulated as optimal control of deterministic or stochastic Navier-Stokes equations. Another problem comes from mathematical finance and is related to option pricing. It includes analysis of infinite dimensional equivalent of ``Black-Scholes" equation and its nonlinear version, so called ``Black-Scholes-Barenblatt" equation. HJB equations in Hilbert (or Banach) spaces are the key to the dynamic programming analysis of optimal control problems of systems driven by partial differential equations.These HJB equations must be investigated from the point of view of generalized solutions. Viscosity solutions should provide the right approach to such equations and the proposed research should be an important ingredient in setting the stage for optimal control of infinite dimensional stochastic systems. The PI also proposes to investigate a class of fully nonlinear non-divergence form uniformly elliptic equations that includes generalizations of quasilinear equations and certain equations of geometric type, an important class in the elliptic theory. Such equations have not been studied systematically, especially when they are discontinuous in the spatial variable. The equations do not have classical solutions and the PI wants to extend the theory of so called L^p-viscosity solutions to this class. In particular the PI plans to investigate the question of regularity of solutions of such equations. This is a major open problem of elliptic partial differential equations and the PI proposes several possible new approaches to it that may give rise to new and interesting techniques.The notion of viscosity solution is one of the main tools of nonlinear partial differential equations and it has found applications in areas as diverse as optimal control, image processing, moving fronts and phase transitions, statistical mechanics, economics, mathematical finance. The motivation for studying some problems described in the proposal comes from optimal control,especially control of stochastic partial differential equations. Their theory is in a state of rapid development and is fueled by modeling questions coming from physics, population biology, chemistry, and economics and mathematical finance. The problem of optimal control of fluid flow is one of the basic engineering problems and has numerous applications in areas like combustion theory, aero and hydrodynamic control, Tokomak magnetic fusion, ocean and atmospheric prediction just to name a few. Problems related to the Musiela model of interest rates come from the modern theory of option pricing. The research of the project should contribute to the development of new directions in partial differential equations and should also have impact on the applied areas mentioned above.