Global Solutions and Ill-Posedness for the Kaup System and Related Boussinesq Systems

The two-way propagation of a certain class of long-crested water waves is governed approximately by systems of Boussinesq-type equations. First introduced by Boussinesq in the 1870s, these equations have been put forward in various forms by many authors. Considered here is a class of such systems which includes the well-known one first introduced by Kaup. The Kaup system is especially interesting since it features an associated inverse scattering formalism, which means that quite detailed aspects of its solutions may be within reach. However, this system and others like it were called into question in earlier work because the initial-value problems for their linearizations around the rest state are ill posed. It is here shown that nonlinearity does not erase this problem. That is to say, the initial-value problem for the Kaup system and others in a certain class of Boussinesq-type systems are ill posed in Sobolev spaces. Indeed, it is shown that arbitrarily small, smooth solutions can blow up in arbitrarily short time in Sobolev-space norms. This norm-inflation result indicates the system is not a good candidate for use in practical problems.