This week at AIM features a MPE related workshop concerned with boundary-value problems for nonlinear dispersive evolution equations and systems. The workshop is organized by Jerry Bona, Min Chen, Shuming Sun, and Bingyu Zhang and has participants with diverse interests in both the pure and applied aspects of such problems.
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 workshop is to advance the study of boundary-value problems for nonlinear dispersive wave equations. Some specific topics that are being considered are:
- Investigating the smoothing properties enjoyed by solutions of boundary-value problems and associated well-posedness theory.
- Investigating 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.
- Extending the theory to multi-space dimensional problems arising in geophysical applications such as coastal dynamics and elsewhere.