Monotonicity formula methods for nonlinear PDEs

已知对于二阶线性椭圆型和抛物型方程，可以由解构造单调函数。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，平均值积分是球半径的单调函数。有更复杂的例子，这种排序，如Almgren的频率公式，其中除其他外，有助于确定结构的零套谐波函数。本项目的目的是构造一些非线性方程的解的单调函数。这种类型的算子的选择是足够的，因为有各种物理问题的非线性方程出现。例如，剪切张量与速度的幂律关系的非牛顿流体的流动，湍流状态下多孔介质中的气体流动，量子场论和两个生物群体的相互作用没有自限性。我们的目的是构造单调函数的三个自由边界问题的非线性控制方程，并指出一些应用在随机博弈论（拔河模型），化学动力学和燃烧（香烟的阴燃和火焰传播）。

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).