Long-time Behavior of Some Dispersive and Fluid Equations

University Of Southern California, Los Angeles CA

Investigators

Abstract

This project is devoted to the study of several partial differential equations that arise in mathematical physics and fluid dynamics. We attempt to make progress on these problems by combining different ideas from different subjects of mathematics. The first problem concerns the motion of the free water surface in the presence of a curved bottom topography. As a first step towards understanding the complex dynamics of waves in the ocean, we will consider small perturbations to the flat water surface and analyze how such perturbations propagate over long time periods. The second problem is about the long-time phenomenon associated with the Schrodinger?s equation in a periodic box. We intend to demonstrate how the fine number theoretic properties of the shape and the size of the box will affect the long-time behavior of Schrodinger waves. The third problem involves the magnetohydrodynamics (MHD) equations, which describes many important plasma systems such as the sun. In particular, in the presence of a strong background magnetic field, we will study the formation of Alfven waves in the vanishing viscosity limit. The fourth and last problem is related to the phenomenon of inviscid damping in fluid dynamics, which is unstable and requires some subtle conditions on the fluid. We extend our previous work on this topic, and intend to study when and how inviscid damping may fail, provided that these conditions are violated. There are specific difficulties in each of the four problems that require a novel approach. In the first problem, the equation involved is a nonlinear dispersive equation with variable coefficients, so the existing methods which have been successful in treating constant coefficient dispersive equations are no longer applicable. Nevertheless, we plan to combine this method, which is based on Fourier space analysis, with appropriate physical space methods to study this problem. In the second problem we are considering Schrodinger?s equation on non-rectangular tori that lack tensor product structure. For this reason we will bring in more advanced techniques from number theory and harmonic analysis, such as the decoupling methods, and counting formulas for solutions to Diophantine systems. The third problem involves a weakly dissipative system which in the vanishing viscosity limit becomes dispersive, and thus we will need to extend the methods that are suitable for dispersive equations to the dissipative context, which on the Fourier side amounts to combining dissipation decay with stationary phase. In the fourth problem, the main challenge is extending the linear analysis in our previous work to infinite time, which requires a combination of Fourier space and physical space techniques (for example, energy estimates with exponential weights both in Fourier and in physical space). 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.