Analytical and geometrical properties of non linear diffusion equations

该研究项目主要集中在扩散过程的各个方面。扩散的数学概念是试图量化和建模物质、流体、热量、粒子或信息如何由于压力的影响（如粒子或种群）或其他邻居与邻居的相互作用（如信息动力学）而及时传播。数学上描述扩散的方法之一是通过偏微分方程，它模拟了无穷小的相邻相互作用。 随着生物学、金融学和社会科学等领域的数学建模，出现了理解扩散过程考虑远程信息或相互作用的现象的需要;即粒子被传输，信息在许多尺度上同时通信，生物体通过化学势的创建进行通信，或者股票以不连续的方式改变价值。PI将研究与这些过程相关的各种现象，这些过程在空间和时间（记忆）中具有长时间的相互作用，例如随着时间堵塞的水库中的流动，发生在远处的隔离过程，或买方和卖方之间存在差距的价格形成模型。第一个研究领域包括涉及空间和时间非局部性的非线性问题。从随机方面来看，该模型是时间上连续的随机游走方程，其中包含Levy游走而不是跳跃。从变分的角度来看，有势压力的多孔介质流动有多种模型，其中介质被流动变形。这些涉及研究完全非线性方程的非局部型，其性质的不变性质平行方程涉及对称函数的海森，如蒙赫-安培方程。 另一个研究领域涉及相变和自由边界问题。一组涉及模型的种族隔离，最佳分区的一个域不相交的子域优化一些“形状”的价值函数。另一组涉及均匀化的前在随机或周期性的媒体，和第三个关注的规则性的自由边界的一些固定或发展的问题。上述问题是普遍的意义上说，同样的范式再次出现在几何和分析，流体动力学和材料科学，金融数学，以及最近的生物学和随机几何。

This research project is focused mainly on aspects of diffusion processes. The mathematical idea of diffusion is an attempt to quantify and model how a species, a fluid, heat, particles, or information spreads out in time due to the effect of pressure (as in particles or populations) or by other neighbor-to-neighbor interaction, as in the case of information dynamics. One of the ways in which diffusion has been described mathematically is through partial differential equations, which model infinitesimal adjacent interactions. With mathematical modeling in fields like biology, finance, and the social sciences, the need has emerged of understanding phenomena where the diffusion process takes into consideration long-range information or interactions; that is the case when particles are transported, information is communicated simultaneously at many scales, organisms communicate by the creation of a chemical potential, or stocks change value in discontinuous ways. The PI will study diverse phenomena related to these processes with long interactions in space and time (memory), such as flows in reservoirs that clog with time, segregation processes that occur at a distance, or models in price formation where there is a gap between buyers and sellers.A first area of research encompasses nonlinear problems involving nonlocality in both space and time. From the stochastic side, the model is the continuous in time random walk equation, which involves Levy walks instead of jumps. From the variational side, there are diverse models for porous medium flows with potential pressures, where the medium is deformed by the flow. These involve study of fully nonlinear equations of nonlocal type that by the nature of their invariant properties parallel equations involving symmetric functions of the Hessian, such as the Monge-Ampere equation. Another area of investigation involves phase transitions and free boundary problems. One group concerns models for segregation of species, optimal partition of a domain by disjoint subdomains optimizing some "shape" value function. Another group deals with the homogenization of fronts in random or periodic media, and a third concerns the regularity of free boundaries for some stationary or evolution problems. The issues described above are universal in the sense that the same paradigm reappears in geometry and analysis, fluid dynamics and material sciences, financial mathematics, and more recently biology and stochastic geometry.