Some Dynamical Questions in Hamiltonian Partial Differential Equations

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

For many physical phenomena dissipation of energy effects are very small and ignoring them could provide a good approximation. Such phenomena include large scale water waves in the ocean, the plasmas in space and controlled fusion devices, dynamics of stars and galaxies in astrophysics. The dynamical behavior of mathematical models of these phenomena is complex and includes formation of coherent structures, such as tsunamis in the ocean, rotating stars (e.g. Sun) and the observed galaxy structures, to name a few. One of the objectives of this project is to contribute to the understanding of the formation and persistence of such coherent structures observed in experiments and in nature. Another topic is to understand and control the instability in many applications. One such example is to control the instability of plasmas in fusion devices to achieve the goal of controlled nuclear fusion for energy production. Another example is the instability of radially rotating neutron stars, which is related to the detection of gravitational waves. Methods of mathematical analysis are the primary tools employed in this investigation. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies. The project will incorporate research of students from graduate through postdoctoral levels thus providing for a vigorous mentoring program. Many conservative models have Hamiltonian structures. One goal of this proposal is to find an instability index formula for Hamiltonian PDEs with an indefinite energy functional, including gravity water waves, ion acoustic wave equations and Vlasov models for collisionless plasmas. The second goal is to find stability criteria (particularly turning point principles) for several stellar models including rotating and magnetic stars, neutron stars and relativistic galaxies. The third goal is to understand the long-time dynamics near linearly stable Bernstein–Greene–Kruskal (BGK) waves for the one-dimensional Vlasov-Poisson equation. 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.