Document Type
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 and 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.
Publication Date
1-1-2003
Recommended Citation
Bellout, H, S. Benachour and E.S. Titi, "Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations" Nonlinearity 16 (November 2003) 1967-1989.
Original Citation
Bellout, H, S. Benachour and E.S. Titi, "Finite-time singularity versus global regularity for hyper-viscous Hamilton-Jacobi-like equations" Nonlinearity 16 (November 2003) 1967-1989.
Legacy Department
Department of Mathematical Sciences
ISSN
0951-7715
Language
eng
Publisher
Institute of Physics
Comments
This is an Accepted Manuscript of an article published in Music Reference Services Quarterly in 2004, Copyright 2004 by The Haworth Press, Inc., Binghamton, NY.