Mathematical Sciences: Harmonic Analysis Method in Certain Evolution Equations

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Abstract Yang 9532033 The proposed will develop harmonic analysis methods to solve a number of nonlinear evolution equations including the nonlinear Schrodinger equation, the nonlinear wave equation, the modified Korteweg-de Vries equation, the Zakharov systems and a fourth order elliptic equation arising in conformal geometry. In the analysis of the relevant integral operators, harmonic analysis estimates will be used to analyze the oscillatory kernels. The known body of results concerning these equation deal mainly with the equation in the whole space. New ideas and techniques will be needed to handle the problem on domains with boundary. The fourth order equation will require an understanding of the Sobolev inequalities for higher order derivatives. In particular, a search for analogue of the positive mass theorem is required. All of the evolution equations mentioned above represent various models in materials science and fluid mechanics. For example, the collective response of a nearly collisionless plasma when the electron temperature is much greater than the ion temperature is dominated by electron plasma waves (Langmuir waves) and ion acoustic waves. Under appropriate conditions (energy density of the waves small compared to particle thermal energy densities, and characteristic scales large compared to a Debye length), Zakharov's model is a useful description of the nonlinear coupling of these collective models. Schrodinger equation can be understood as a limit case of Zakharov's model and can also be used to describe in certain regimes the Langmuir waves in a plasma, i.e., the time evolution of the envelope of the propagating electric field. Our main project is to analyze the solutions of the equations by using the tools of harmonic analysis. For example, we want to understand the development of singularity at certain time, which represents, in the Schrodinger models, how the Langmuir waves collapse. This typical proble m is understood when the models are on the whole physical space. Part of our work is to study the situation when the models are confined in a bounded region in the physical space. The other project we are working on is to study a non-linear problem in geometry. Here we are looking for a conserved quantity analogous to the total mass of a gravitational systems which will play an important role to the solvability of the equation.