Monotonicity formula methods for nonlinear PDEs

非线性偏微分方程的单调性公式方法

• 批准号: EP/K024566/1
• 负责人: Aram Karakhanyan
• 金额: $ 12.87万
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK024566%2F1
• 关键词: Monotonicity; formula; methods; nonlinear; PDEs

It is known that for linear elliptic and parabolic equations of second order one can construct monotone functions from the solution. A typical example is the mean value integral of harmonic function over a ball. In this case the mean value integral is monotone function of the radius of the ball. There are more complex examples of this sort such as Almgren's frequency formula which, among other things, helps to identify the structure of the zero set of harmonic function. The aim of this project is to construct monotone functions for the solutions of some nonlinear equations. The choice of this type of operators is adequate since there are various physical problems where the nonlinear equations emerge. For instance, the flow of non-Newtonian fluids with power law dependence of the shear tensor from the velocity, the flow of gas in porous media in turbulent regime, the quantum field theory and the interaction of two biological groups without self-limiting. We aim to construct monotone functions for three free boundary problems with nonlinear governing equations and point out some applications in stochastic game theory (Tug-of-War model), Chemical Kinetics and Combustion (smouldering of cigarettes and flame propagation).

已知二阶线性椭圆型和抛物型方程的解可以构造单调函数。一个典型的例子是球上调和函数的均值积分。在这种情况下，均值积分是球半径的单调函数。还有更复杂的例子，比如Almgren的频率公式，它能帮助我们识别谐波函数零集的结构。本课题的目的是为一些非线性方程的解构造单调函数。这种类型的算子的选择是足够的，因为有各种各样的物理问题，其中非线性方程出现。例如，剪切张量与速度幂律相关的非牛顿流体的流动，湍流状态下多孔介质中的气体流动，量子场论以及两种生物群体无自限的相互作用。我们的目标是构造具有非线性控制方程的三个自由边界问题的单调函数，并指出在随机博弈论（拔河模型）、化学动力学和燃烧（香烟的闷烧和火焰传播）中的一些应用。

项目成果

Strongly Nonlocal Dislocation Dynamics in Crystals

• DOI: 10.1080/03605302.2014.914536
• 发表时间: 2013-11
• 期刊: Communications in Partial Differential Equations
• 影响因子: 1.9
• 作者: S. Dipierro;A. Figalli;E. Valdinoci

A class of unstable free boundary problems

一类不稳定自由边界问题

• DOI: 10.2140/apde.2017.10.1317
• 发表时间: 2017
• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Dipierro S

Graph properties for nonlocal minimal surfaces

• DOI: 10.1007/s00526-016-1020-9
• 发表时间: 2015-06
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: S. Dipierro;O. Savin;E. Valdinoci

Boundary behavior of nonlocal minimal surfaces

• DOI: 10.1016/j.jfa.2016.11.016
• 发表时间: 2015-06
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: S. Dipierro;O. Savin;E. Valdinoci

Potential theoretic approach to Schauder estimates for the fractional Laplacian

• DOI: 10.1090/proc/13227
• 发表时间: 2016-02
• 期刊: arXiv: Analysis of PDEs
• 作者: Claudia Bucur;A. Karakhanyan