This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The main thrust of this project is to further the understanding of the global/large data behavior of solutions to certain dispersive equations at critical regularity. More precisely, the principal investigator considers global well-posedness and scattering questions for nonlinear Schrodinger, Klein-Gordon, and (modified) Korteweg-de Vries equations for initial data belonging to critical/low-regularity Sobolev spaces. Critical-regularity problems for the nonlinear wave (NLW) and Schrodinger (NLS) equations have attracted considerable attention over the past few years. These works have developed a powerful set of tools and techniques meant to address NLW and NLS at conserved critical regularity. The main purpose of this project is to strengthen and broaden this toolbox. Immediate goals include treating the focusing (low-dimensional) energy-critical NLS and the defocusing/focusing mass-critical NLS, problems that lie a little beyond the reach of existing techniques (except in the case of radial data). Second, the principal investigator wishes to test the robustness of the toolbox developed thus far against new difficulties, such as problems for which the critical regularity does not correspond to a (coercive) conserved quantity or problems with broken symmetries. The last part of the project is concerned with the global well-posedness question for the (modified) Korteweg-de Vries equation for initial data in low regularity spaces. Thanks to complete integrability techniques, this problem is understood better in the periodic case than in the nonperiodic one. The principal investigator proposes to revisit these new advances due to Kappeler and Topalov from a purely partial differential equations point of view in the hope of discovering an appropriate gauge that would allow the treatment of the nonperiodic case at low regularity.
The equations under investigation in this project have a rich history. They have been studied by mathematicians and physicists alike because they capture important facets of certain physical behaviors, while maintaining an attractive simplicity. As such, they serve as breeding grounds for new analytical techniques for studying partial differential equations. Although the equations to be investigated are drastically oversimplified relative to the needs of science or industry, the principal investigator believes that the study of these equations will foster the development of tools with much broader applicability, while even the tiniest hastening toward an era when supercritical equations such as the celebrated Navier-Stokes equation can be treated would be very beneficial indeed. Parallel to the development of a toolbox is its dissemination. The principal investigator will continue her activities in this direction, including the maintenance of a set of lecture notes on this material.
