Dynamics of Fluid and Nonlinear Waves

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Differential equations are used to model problems arising from a broad background including physics, engineering, biology, finance, etc. Great efforts are made to understand these mathematical models rigorously for a twofold reasons. On the one hand, the validity and the relevance of these ideal models are established through the comparison between the results from theoretical analysis and the experimental data. On the other hand, once the meaningfulness of a mathematical model is supported by available experimental observations to certain degree, the theoretical studies on these ideal models can provide properties and predictions of the original problems that are hard to be obtained through experiments. For systems involving the temporal evolution, of particular interests are those structural and asymptotic properties. These include some special structures, such as steady states, periodic and quasi-periodic solutions, chaotic orbits etc, as well as their qualitative properties like stability etc. 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 extremely important, partly due to the fact that they and some of their associated structures serve as the boundaries separating different collections of stable states in a system. In this project, the PI plans to focus on the local dynamics near steady states in several classical nonlinear partial differential equation systems, which all belong to the general category of nonlinear waves. The lack of a priori damping and the complicated nonlinearity pose most of the challenges in their mathematical analysis. The PI plans to rigorously investigate the local dynamics of the incompressible Euler equation as well as a general class of quasi-linear Hamiltonian partial differential equations. In particular, for the incompressible Euler equation, which models inviscid incompressible fluids such as water, the proposed problems include fluids in rigid containers and fluids with free surfaces like ocean waves. Even though there have been extensive studies on these systems and great progresses have been made in recent year, due to their very complex nature, many issues including some very fundamental ones are still not well understood after years of efforts. The PI plans to focus on their local dynamic structures near equilibria, including stability/instability, local invariant manifolds, special solutions, bifurcations, and singular perturbations. While these aspects are standard notions in the theory of smooth dynamical systems, due to the highly nonlinear nature of these partial differential equations, their solution maps often do not have sufficient smoothness in the infinite dimensional phase spaces for the classical theory to apply directly. In contrast to ordinary differential equations, the relationship between the qualitative structures and the regularity analysis of these nonlinear partial differential equations is an essential analytical aspect of nonlinear partial differential equation dynamics. 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.