Nonhomogeneous boundary-value problems for nonlinear waves

This workshop, sponsored by AIM and the NSF, will be concerned with boundary-value problems for nonlinear dispersive evolution equations and systems.

Nonlinear, dispersive evolution equations and systems of such equations arise as models for wave motion in a very wide variety of physical, biological and engineering. Since the 1960's, there has been a steady increase of interest in the theory and applications of such equations. On the mathematical side, the pioneering work of Ginibre and Velo and Kenig, Ponce and Vega was followed by the spectacular progress of Bourgain, Tao and their collaborators, as well as many others.

If one tries to use the pure initial-value formulations in practice, one is immediately beset by the difficulty of determining accurately a wave profile in the entire spatial domain of its definition at a single instant of time. Generally speaking, this is not possible to accomplish with any semblance of accuracy. Moreover, when these equations are used in engineering and science, the natural way to pose them is with specified, not necessarily homogeneous boundary conditions. And, problems of control of dispersive equations demand a firm grasp of boundary-value problems as a starting point for developing cogent theory.

By contrast with the initial-value problem, theory for boundary-value problems other than those featuring periodicity has generally lagged behind the developments for the pure initial-value problems. The overall goal of the proposed workshop is to advance the study of boundary-value problems for nonlinear dispersive wave equations. Within this larger framework, there are several specific topics we have in mind.

1. Investigate the smoothing properties enjoyed by solutions of boundary-value problems and associated well-posedness theory.
2. Investigate the controllability and stabilizability of solutions of nonlinear, dispersive wave equations. Experience shows that results from the first topic above will be central to such an investigation.
3. Extend the theory to multi-space dimensional problems arising in geophysical applications such as coastal dynamics and elsewhere.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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