The process of diffusion makes a distribution in space become more regular as time goes by. There are many systems that exhibit this behavior. The values of the temperature throughout an object, the concentration of some chemical, or the population density in migration are some examples of quantities that show diffusive behavior. In mathematical analysis, the prime example of a diffusive equation is the heat equation, or the more general family of parabolic partial differential equations. The purpose of this research project is to study non-standard diffusion. The project explores regularization effects in nonlinear equations that are outside the classical framework of parabolic equations. The research studies non-local equations, where the evolution of the value at any point depends on the global picture, and applies those methods in equations from statistical mechanics modeling the density of particles in a dilute gas. The project also studies conservation law equations, a fundamental class of partial differential equation, and aims to explicate regularization properties there as well. The project will involve graduate students and postdoctoral associates in the research. Results will be disseminated through publications, conferences, and summer schools.
The study of non-local equations, and more precisely integro-differential equations, is a very active area of research. Recent work indicates that some of the ideas and methods developed in that area can be used to obtain regularity estimates for the Boltzmann equation from statistical mechanics. This research project aims at proving that if the solution of the Boltzmann equation ever develops a singularity, then some hydrodynamic quantity must be unbounded. These hydrodynamic quantities are the mass, energy, and entropy density, which correspond to physically meaningful properties of the fluid. Thus, the project aims to establish that if there is a singularity in the mesoscopic description of the fluid given by the Boltzmann equation, it must be apparent macroscopically. The Boltzmann equation can be rewritten as a parabolic integro-differential equation, and the restriction on its hydrodynamic quantities determines the non-degeneracy of the corresponding kernel. This connection between integro-differential equations and the Boltzmann equation is also useful as a guide for further development of the former subject. The study of regularization mechanisms using methods from parabolic equations applied to conservation law equations is arguably even more surprising. In scalar conservation laws, the solution is transported following a velocity that depends on the value at every point. Shock singularities are unavoidable for smooth initial data; however, the same shock mechanism forces some self-organization that translates into a regularization effect when starting from very rough initial data.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
