Journal article
Global Solutions and Ill-Posedness for the Kaup System and Related Boussinesq Systems
Indiana University mathematics journal, v 68(4), pp 1173-1198
01 Jan 2019
Featured in Collection : UN Sustainable Development Goals @ Drexel
Abstract
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.
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Details
- Title
- Global Solutions and Ill-Posedness for the Kaup System and Related Boussinesq Systems
- Creators
- David M. Ambrose - Drexel Univ, Dept Math, Philadelphia, PA 19104 USAJerry L. Bona - Univ Illinois, Dept Math Stat & Comp Sci, Chicago, IL 60607 USATimur Milgrom - E Trade Financial Corp, 671 N Glebe Rd, Arlington, VA 22203 USA
- Publication Details
- Indiana University mathematics journal, v 68(4), pp 1173-1198
- Publisher
- Indiana Univ Math Journal
- Number of pages
- 26
- Grant note
- DMS-1515849 / National Science Foundation; National Science Foundation (NSF) Ulsan National Institute of Science and Technology Universite Paris-Est Creteil National Center for Theoretical Science of the National Taiwan University
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000484366600005
- Scopus ID
- 2-s2.0-85071756610
- Other Identifier
- 991019169146904721
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- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics