论文标题
多孔介质方程和反应扩散系统的拉格朗日解决方案
Lagrangian solutions to the Porous Media Equation and Reaction Diffusion Systems
论文作者
论文摘要
在本文中，我们沿着多孔介质方程式弱解决方案产生的压力梯度构建了全球向后和向后的拉格朗日流图。主要困难是，当初始数据具有紧凑的支持时，众所周知，压力梯度不是BV函数。因此，常规拉格朗日流的理论不能应用于构建流图。为了克服这一难度，我们开发了一个新的论点，将Aronson-Bénilan类型估计与Crippa和de Lellis的定量拉格朗日流理论相结合，以表明某些双重对数量衡量流量图的稳定性并不能足够快地吹出以防止紧凑的紧凑。我们的论点足够灵活，可以处理Hele-shaw限制和多孔介质方程的多种概括，在该方程中，该方程被反应扩散方程的耦合双曲线 - 促糖系统代替。作为流图的一种应用，我们能够构建解决方案，如果不同物种在初始时间分离，则不能将它们混合在一起。
In this paper, we construct global-in-time forward and backward Lagrangian flow maps along the pressure gradient generated by weak solutions of the Porous Media Equation. The main difficulty is that when the initial data has compact support, it is well-known that the pressure gradient is not a BV function. Thus, the theory of regular Lagrangian flows cannot be applied to construct the flow maps. To overcome this difficulty, we develop a new argument that combines Aronson-Bénilan type estimates with the quantitative Lagrangian flow theory of Crippa and De Lellis to show that certain doubly logarithmic quantities measuring the stability of flow maps do not blow up fast enough to prevent compactness. Our arguments are sufficiently flexible to handle the Hele-Shaw limit and a multispecies generalization of the Porous Media Equation where the equation is replaced by a coupled hyperbolic-parabolic system of reaction diffusion equations. As one application of our flow maps, we are able to construct solutions where different species cannot mix together if they were separated at initial time.
