Analytical and geometrical properties of non linear diffusion equations

本研究项目主要集中在扩散过程的各个方面。扩散的数学思想是试图对物种、流体、热量、粒子或信息如何由于压力(如粒子或种群)或其他邻居之间的相互作用的影响而及时扩散进行量化和建模，如信息动力学的情况。数学描述扩散的方法之一是通过偏微分方程式，它模拟了无限小的相邻相互作用。随着生物、金融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；在这种情况下，粒子被运输，信息在多个尺度上同时传递，生物体通过创造化学势进行交流，或者股票以不连续的方式改变价值。PI将研究与这些在空间和时间(记忆)上长时间相互作用的过程相关的各种现象，例如被时间堵塞的水库中的流动，发生在远处的分离过程，或者买家和卖家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动方程，其中包含Levy游动而不是跳跃。从变分的角度来看，对于具有势压的多孔介质流动，有不同的模型，其中介质被流动变形。这涉及到研究非局部类型的完全非线性方程，根据其不变性质，平行的涉及黑森对称函数的方程，如Monge-Ampere方程。另一个研究领域涉及相变和自由边界问题。一组涉及物种分离的模型，通过不相交的子域优化某些形状值函数来优化区域的最优划分。另一组研究随机或周期介质中波面的均匀化，第三组研究一些静止或演化问题的自由边界的正则性。上述问题具有普遍性，因为在几何学和分析、流体力学和材料科学、金融数学以及最近的生物学和随机几何学中，都出现了相同的范式。

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.