The study of partial differential equations of three types motivates this work, which combines methods of harmonic and classical analysis in treating the problems outlined in the research plan. It is well known that the free Schrodinger operator defined on Euclidean space of any dimension maps integrable functions into bounded functions whose bound decays with reciprocal powers of time. Global existence for some non-linear Schrodinger equations with small initial data as well as results in scattering theory derive from this fact. The estimate is poorly understood and allows no flexibility for applications to the case where a potential is added to the Laplace operator. Very recent results have now resulted in positive and sharp estimates on the operator-plus-potential. However, the conditions required of the potential are too restrictive to include physically significant examples. This work will continue efforts to determine the full set of potentials for which the estimates actually hold. A second line of investigation concerns the Euler equation for an inviscid and incompressible fluid flow. When incompressible fluid flow occurs with a high Reynolds number (a measure of viscous resistance) in 3-dimensions, singularities form in finite time. However, this has only been demonstrated numerically. Because the singularities can only occur if the vorticity is stretched, numerical approximation of solutions which form singularities is very subtle. In this project a rigorous study of some prototype models with exact solutions will be made. These will then be used to check numerical methods and possibly design new ones if the available ones are not sensitive enough. The third problem area concerns operators arising in the study of partial differential equations with rough coefficients. The focus is on a question which has become known as the square- root Kato conjecture. It concerns the natural domain of the square-root of a divergence type second order differential operator. This domain is believed to be a specific Sobolev space of functions. A discovery by A. McIntosh led to a deep connection between the identification of this domain and the boundedness of Cauchy-kernal operators on Lipschitz curves, a problem of considerable interest, which harmonic analysts have been studying for nearly two decades. Recent work suggests that a resolution of the conjecture may be at hand - at least in spaces of dimension less than five. Work will continue toward this goal.
