Fred Weissler, University Sorbonne Paris Nord

Abstract: After rapidly reviewing the history of the Korteweg-deVries equation, I will discuss the issues of local and global well-posedness for complex-valued solutions on the circle. In particular, a large class of local-in time solutions develop a singularity in finite time. Hence global well-posedness can fail.
In addition, local well-posedness can fail for two reasons.  There exist initial values with no reasonable local-in time solution. Also, if complex-valued solutions are considered, continuous dependence fails at the zero solution.  These results are of course completely at variance with the well-known results for real-valued solutions.