Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations Academic Article

abstract

• The global regularity for the two- and three-dimensional Kuramoto-Sivashinsky equations is one of the major open questions in nonlinear analysis. Inspired by this question, we introduce in this paper a family of hyper-viscous Hamilton-Jacobi-like equations parametrized by the exponent in the nonlinear term, p, where in the case of the usual Hamilton-Jacobi nonlinearity, p = 2. Under certain conditions on the exponent p we prove the short-time existence of weak and strong solutions to this family of equations. We also show the uniqueness of strong solutions. Moreover, we prove the blow-up in finite time of certain solutions to this family of equations when the exponent p > 2. Furthermore, we discuss the difference in the formation and structure of the singularity between the viscous and hyper-viscous versions of this type of equation.

authors

• Titi, Edriss

published proceedings

• NONLINEARITY

author list (cited authors)

• Bellout, H., Benachour, S., & Titi, E. S.

citation count

• 21

complete list of authors

• Bellout, H||Benachour, S||Titi, ES

publication date

• November 2003

publisher

• IOP Publishing  Publisher

published in

• NONLINEARITY  Journal

keywords

• 49 Mathematical Sciences
• 4901 Applied Mathematics
• 4904 Pure Mathematics

Digital Object Identifier (DOI)

• 10.1088/0951-7715/16/6/305

start page

• 1967

end page

• 1989

volume

• 16

issue

• 6

URL

• http://dx.doi.org/10.1088/0951-7715/16/6/305