The project is devoted to the study of convection-dominated linear and nonlinear parabolic PDEs that arise in various contexts in mathematical physics. It is aimed at answering some outstanding questions concerning their asymptotic behavior and a better understanding of the mechanisms behind the pattern formation associated with these equations. In particular, one of the central goals is to provide rigorous justifications for both the pattern formation observed in fluid mixing--the so-called 'strange' eigenmodes, as well as for the existence of kinematic dynamos. The investigator takes two distinct approaches; the first one is to extend the existing theory of principal Floquet bundles and exponential separation to a class of non-autonomous linear equations featuring advection by velocity fields with periodic and general time-dependence and subject to more physical boundary conditions. The second more ambitious approach is via non-autonomous inertial manifolds, which provide an avenue for a complete decomposition in Floquet bundles. While the theory of attractors is now very well developed, the theory of inertial manifolds is lagging in many respects. The question of existence of inertial manifolds for many physically relevant parabolic PDEs, including the Navier-Stokes equations, for which most of the theory was originally developed, is still open. The PI investigates the possibility of extending the transformation method, which he recently developed for a class of viscous Burgers equations and a class of nonlinear Fokker-Planck equation to a larger class of convection-dominated parabolic equations that fail to satisfy the spectral-gap condition. The transformation method exploits deeper and more subtle interplay between the diffusive and convective terms, which has been largely ignored in other approaches so far.
The research project lies at the crossroads of several different fields of mathematical physics, ranging from global climate and ocean dynamics, fluid mixing and turbulence to flame propagation and polymer dynamics. It employs methods from the theory of partial differential equations and the theory of infinite and finite-dimensional dynamical systems. Many interesting and important physical phenomena are modeled by convection-dominated parabolic partial differential equations which feature both a diffusive term that has an entropic effect and leads to irreversible loss of structure, as well as convective terms that lead to more organized states. These two mechanisms are not separate, but act rather in unison, leading to pattern formations, whose understanding is one of the central problems of mathematical physics. This project is aimed at identifying and exploiting some subtle mechanisms of the interplay between diffusion and advection in order to mathematically rigorously explain the formation and properties of these patterns. Results of this research have immediate practical implications. For example, they lead to understanding of how parcels of fluid move and mix, which has an impact in climatology, pollution dispersion and marine ecology.
The advection-diffusion equation constitutes an important paradigm for a wide range of physical, chemical, and biological processes that are characterized both by transport induced by a fluid flow as well as small diffusion of different nature. Examples include homogenization in fluid mixtures, pollutant dispersion in the ocean or atmosphere, distribution of ozone in the atmosphere, polymer dynamics in flowing media, temporal evolution of biological systems in flowing media, energy transport in flowing media, etc. Applications of practical importance range from micro-mixers and internal combustion engines to global ocean and climate dynamics. A large area of active scientific inquiry remains the rather intricate interplay between diffusion and advection in the limit for vanishing diffusivity. While to the layman’s eyes advection and diffusion may appear as two competing processes (diffusion leads to entropy, while advection creates structure), they are actually cooperative, and the result is enhanced rates of dissipation, relaxation or mixing, depending on the situation. It was found that certain flows have the ability to significantly enhance the mixing properties of the fluid. In the absence of advection under diffusion alone, mixing occurs over timescales that are inversely proportional to the vanishing diffusivity. Certain flows, however, enhance the mixing (dissipation, relaxation), which can occur over much shorter convective timescales. This phenomenon has been a subject of scientific inquiry for some time. The majority of the existing mathematical literature, however, is devoted to the understanding of the flows that are dissipation enhancing. For the majority of the situations of interest in ocean and atmosphere dynamics, this is not the case. The main contribution of the project "Floquet bundles and inertial manifolds for convection-dominated parabolic PDEs" is a result about advection-driven enhancement of the mixing for the flows that are not dissipation enhancing. While the dissipation still occurs over longer diffusive timescales, some aspects of the mixing occur over shorter convective timescales resulting in the creation of patterns that have long intrigued experimental physicists. During the duration of the project, the problem was tackled both numerically and analytically. The project had an educational impact in that students were involved in various aspects of the investigation, especially the computational ones. Some results became parts of students’ dissertation theses.
