Nonlinear partial differential equations and applications

本文建议发展研究非线性、双曲型和抛物型/椭圆型偏微分方程解的一般方法。更准确地说，利用分析、偏微分方程和概率论的技术，PI计划继续他的程序在以下四个一般领域工作：a.一阶和二阶完全非线性随机偏微分方程组。B.相变。C.湍流反应扩散方程和燃烧。一阶和二阶完全非线性、退化、椭圆型和抛物型方程。在这个建议中考虑的大多数偏微分方程式都是作为连续介质和统计物理的模型出现的。这些模型出现在各种主题中，从材料科学和相变(锋面运动、介观和宏观尺度、均化)，到流体流动(湍流反应-扩散和燃烧)，以及随机分析(相互作用的粒子系统、随机环境中的流动和随机速度，以及随机控制)。对这些模型的定性分析有助于更好地理解实际的物理问题，并在许多情况下为开发高效的数值算法提供了基础。

This is a proposal to develop general methods to study nonlinear, hyperbolic and parabolic/elliptic partial differential equations. More precisely using techniques from analysis, partial differential equations and probability, the PI plans to continue his program working in the following four general areas: A. Fully nonlinear stochastic partial differential equations of first-and second- order . B. Phase Transitions. C. Turbulent reaction-diffusion equations and combustion. D. First- and second-order fully nonlinear, degenerate, elliptic and parabolic equations. Most of the partial differential equations considered in this proposal arise as models in continuum and statistical physics. These models appear in a variety of topics ranging from material sciences and phase transitions (motion of fronts, mesoscopic and macroscopic scales, homogenization), in fluid flows (turbulent reaction-diffusion and combustion), and in stochastic analysis (interacting particle systems, flows in random environments and with random velocities, and stochastic control). The qualitative analysis of these models contributes to the better understanding of the actual physical problems, and provides, in many cases, the foundation for the development of efficient numerical algorithms.