Analysis of Fluids and Nonlinear Waves

Partial differential equations (PDE) are widely used to model various problems involving spatial and temporal variables arising in physics, engineering, biology, finance, etc. The aims of the efforts to understand rigorously these mathematical models are twofold. On the one hand, the physical relevance and the validity of these ideal models are established through the comparison between the results from theoretical analysis and the experimental observations. On the other hand, once the meaningfulness of a mathematical model is supported by available experimental data to certain extent, the theoretical studies on these ideal models can provide properties and predictions of the original physical problems that are difficult to obtain from experiments. For physical systems involving temporal evolution, of particular interest are certain structural and asymptotic properties. These include special structures, such as equilibria, periodic and quasi-periodic orbits, chaotic orbits, and their qualitative properties like stability or asymptotic stability. In general, on the one hand, only stable states are physically observable in a system, while the ideal, but unstable, states are hardly observed due to their extremely sensitive dependence on the parameters. On the other hand, unstable states are also very important, in part because they and some of their associated structures serve as the boundaries separating different collections of stable states in a system. In this project, the principal investigator (PI) plans to focus on the local dynamics near steady states in several classical nonlinear PDE systems which belong to the general category of nonlinear waves and incompressible fluids. The complicated nonlinearity poses tremendous challenges in their mathematical analysis. A substantial part of the project is suitable for graduate students and postdocs and provides research training opportunities for these early-career mathematicians. More specifically, the project will study the dynamics of incompressible fluid PDE (inviscid, weakly viscous, or with density stratification) with free surfaces as well as a class of nonlinear Hamiltonian PDE. They are standard models arising in fluids, atmosphere-oceans, nonlinear waves, etc. Their solution flows generate infinite dimensional dynamical systems in function spaces. There has been extensive research on these systems with many important advances in recent years. However, due to the complicated spectra of the linearized problems, the highly nonlinear nature, regularity issues, and the multiple scales in space and time they involve, many questions, including some fundamental ones, are still not well understood. First, the PI will work on the two-dimensional water waves linearized at shear flows, including the bifurcation of instability and the linear inviscid damping for the gravity water waves and the spectra and linear flows of the stratified water waves. The second focus of the project is the nonlinear local dynamics of a class of Hamiltonian PDE including the local invariant manifolds for quasilinear Hamiltonian PDE, where the regularity issue poses a major challenge, and the unfolding bifurcation of small homoclinic type solutions in a singular perturbation framework. The PI will also study a potential flow approximation to weakly viscous water waves including formal justification via detailed multi-scale expansions involving boundary layers followed by rigorous proofs. Understanding and solving these problems, expected to be largely based on their specific mechanical and geometric structures, would result in substantial theoretical advances in these areas and possibly lead to the discovery of new physical and mathematical phenomena in the underlying systems. 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. NSF Award ID: 2350115 | Program: 01002425DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Chongchun Zeng | Institution: Georgia Tech Research Corporation, ATLANTA, GA | Award Amount: $299,998 View on NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350115 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2350115.html