Analytical and geometrical properties of non linear diffusion equations

University Of Texas At Austin, Austin TX

Investigators

Abstract

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.