In many physical problems, the dissipation is very small and conservative models can be used for their study. These include the fluid motions in atmosphere and ocean, the Bose-Einstein condensate in quantum mechanics, the plasmas in controlled fusion devices, etc. The dynamical behavior of these models is very complex due to the lack of strong dissipation. One important topic is to understand the formation and persistence of coherent structures observed in experiments and in nature. Some examples include the large vortex structures in the atmosphere and ocean, the traveling wave patterns in ocean waves, and the observed galaxy structures. 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. Methods of mathematical analysis are the primary tools employed in the proposed investigations. The rigorous mathematics makes it feasible to do stable numerical computations and to better understand the phenomena found in numerical and experimental studies.
One focus of the project is to understand invariant structures including quasi-periodic solutions and invariant manifolds, near physically interesting equilibria of several PDE models, including Vlasov equations, incompressible Euler and Naiver-Stokes equations, and Gross-Pitaevskii equation. These invariant structures form a skeleton in the physical phase space and provide important clues to study long time dynamical behaviors. Topics to be studied include: the regularity threshold for nonlinear damping and the existence of nontrivial invariant structures near simpler equilibria of non-dissipative models including Vlasov models and 2D Euler equations, the construction of invariant manifolds in many conservative continuum models and Hamiltonian PDEs, and the persistence of invariant structures for large time scale under small dissipation. Also, new approaches will be developed to find stable BGK waves and electrostatic solitary waves of Vlasov-Poisson systems.
