Dynamics of inviscid fluids and nonlinear waves

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Partial differential equations are widely used to model various problems arising from physics, engineering, biology, finance, etc. The aims of the efforts to understand these mathematical models rigorously 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 of the original physical problems that are hard to be obtained through experiments. For physical systems involving temporal evolution, of particular interests are those structural and asymptotic properties. These include some special structures, such as steady states, periodic and quasi-periodic orbits, chaotic orbits etc, and 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 very 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. More specifically, the PI proposes to study rigorously the local dynamics of the following partial differential equation systems. The first one is the incompressible Euler equation, which models non-viscous and incompressible fluids like water. The proposed problems include fluids in rigid containers and fluids with free surfaces like ocean waves. The second partial differential equation in the proposal is the Vlasov-Poisson system that models the collisionless plasma, which consists of particles with both velocity and electrical charge. The third one is the quasi-linear waves equation. Even though there have been extensive research on these systems and many important progresses have been made in recent year, due to their very nonlinear nature, many issues including some fundamental ones are still not well understood after years of efforts. The PI plans to focus on their local dynamic structures near equillibria, including stability/instability, local invariant manifolds, special solutions and bifurcations. 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 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. Solving and understanding 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 corresponding systems.