Hamiltonian Methods for Dispersive Fluids and Plasmas

Brown University, Providence RI

Investigators

Abstract

Physical phenomena related to galaxies, gases, and plasmas can be modeled by the partial differential equations describing fluid motions. This project addresses several aspects of such equations, including the qualitative and quantitative study of their asymptotic behavior and the stability of various equilibriums (stable equilibriums correspond to objects that can be encountered/observed). An underlying theme of the project is to develop robust methods to leverage the stabilizing mechanisms of dispersion (the fact that, since all objects move constantly in an infinite space, a high concentration at any given point at any given time becomes increasingly unlikely as time passes). Another central theme of the project is to study the effect of rotation on fluids. While it has been already recognized that this induces a stabilizing dispersive effect, the extent of this effect, its precise mathematical expression, and its consequences, remain largely unknown. This project will also contribute to preparing the next generation of scientists by training graduate students, developing courses, as well as collaborative opportunities. This project addresses various stability and asymptotic questions on kinetic and fluid equations, using the dispersive mechanism to obtain long-time control of the solutions. For kinetic equations, a goal is to understand the long-time behavior of solutions with a point charge part, using (and developing) the method of asymptotic action-angle. A first example is the stability of a Dirac mass, in the repulsive or (tentatively) the attractive case. A second question is to investigate the case of large data and the extension to the relativistic case. Another problem addressed concerns homogeneous equilibrium with an aim to better understand Landau damping (in the whole space). The PI will consider a model of fat-tail equilibria (the Poisson equilibrium) and will prove stability, by decomposing the electric field into an electrostatic contribution with faster decay and an oscillatory component that dissipates slowly. Using normal form techniques, the PI will prove that the long-time contribution of the slow but oscillatory component remains under control, and will close with a bootstrap argument based on Lebesgue norm of the density alone. Other generalizations will also be pursued. Finally, this project also considers situations where the rotation induces a stabilizing decaying mechanism, starting with global stability of the vector field of rigid motion for the incompressible 3d Euler. This in turn reduces to a problem of small data global existence for a quasilinear dispersive problem, which will be tackled using methods originating from the space-time resonance method and extensions. 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.