Coherent Structures and Nonlinear Partial Differential Equations

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Coherent structures, such as the elliptical or spiral galaxies in the universe, or large scale vortex structures in the atmosphere and ocean, or surface water waves, are observed in many natural phenomena. This research is aimed at understanding the formation and evolution of such coherent structures (that in some cases are amenable to mathematical analysis). A first question is their stability, that is, whether they persist even if they experience perturbations. Understanding the mechanisms of instability and stability is fundamental for many applications. An example is the importance in plasma physics and engineering of understanding mechanisms of instability, to enable design of devices able to contain plasma long enough for practical fusion energy production. Another topic is to understand the dynamical roles of coherent structures. For example, how do we explain the appearance of the large-scale structures observed in the atmosphere and oceans? What is the mechanism for an initially unstructured state to approach a final coherent state? Such problems are far from being thoroughly understood. 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.One focus of the project is the stability problem for Hamiltonian partial differential equations with energy functional of infinite Morse index. They include gravity water waves (and many long wave models), nonlinear Dirac equations, ion acoustic wave equations, and Vlasov models for collisionless plasmas. In contrast to the cases with energy functional of finite Morse index, to date there are very few general tools for stability problems with infinite Morse index. Several models, including the regularized Boussinesq equation and two-dimensional Euler equation, will be studied, with the goal to develop some general tools for stability problems of infinite Morse index. Another focus of the project is the metastability of coherent structures and the persistence of invariant structures under small dissipation. They include the metastability of quasi-stationary flows of the two-dimensional Navier-Stokes equation with small viscosity, and the inviscid limit of invariant manifolds of Navier-Stokes equations. Hamiltonian and geometric structures of the fluid equations will be explored in these investigations.