Lipschitz continuous solutions of some doubly nonlinear parabolic equations

Mitsuharu Ôtani, Yoshie Sugiyama

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W01,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L-energy method", which provides a main tool for our arguments and would be useful for other situations.

Original languageEnglish
Pages (from-to)647-670
Number of pages24
JournalDiscrete and Continuous Dynamical Systems
Volume8
Issue number3
Publication statusPublished - 2002
Externally publishedYes

Fingerprint

Continuous Solution
Nonlinear Parabolic Equations
Lipschitz
Porous materials
Porous Medium Equation
Local Solution
Non-Newtonian Fluid
Energy Method
P-Laplacian
Mean Curvature
Classical Solution
Degeneracy
Global Solution
Filtration
Evolution Equation
Weak Solution
Fluids
Non-negative
Singularity
Term

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Analysis
  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Lipschitz continuous solutions of some doubly nonlinear parabolic equations. / Ôtani, Mitsuharu; Sugiyama, Yoshie.

In: Discrete and Continuous Dynamical Systems, Vol. 8, No. 3, 2002, p. 647-670.

Research output: Contribution to journalArticle

Ôtani, Mitsuharu ; Sugiyama, Yoshie. / Lipschitz continuous solutions of some doubly nonlinear parabolic equations. In: Discrete and Continuous Dynamical Systems. 2002 ; Vol. 8, No. 3. pp. 647-670.
@article{de5b546812b6429e8370d64a9a151344,
title = "Lipschitz continuous solutions of some doubly nonlinear parabolic equations",
abstract = "This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W01,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method {"}L∞-energy method{"}, which provides a main tool for our arguments and would be useful for other situations.",
author = "Mitsuharu {\^O}tani and Yoshie Sugiyama",
year = "2002",
language = "English",
volume = "8",
pages = "647--670",
journal = "Discrete and Continuous Dynamical Systems",
issn = "1078-0947",
publisher = "Southwest Missouri State University",
number = "3",

}

TY - JOUR

T1 - Lipschitz continuous solutions of some doubly nonlinear parabolic equations

AU - Ôtani, Mitsuharu

AU - Sugiyama, Yoshie

PY - 2002

Y1 - 2002

N2 - This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W01,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.

AB - This paper is concerned with two types of nonlinear parabolic equations, which arise from the nonlinear filtration problems for non-Newtonian fluids. These equations include as special cases the porous medium equations ut = div(ul∇u) and the evolution equation governed by p-Laplacian ut = div(|∇u|p-2∇u). Because of the degeneracy or singularity caused by the terms ul and |∇u|p-2, one can not expect the existence of global (in time) classical solutions for these equations except for special cases. Therefore most of works have been devoted to the study of weak solutions. The main purpose of this paper is to investigate the existence of much more regular (not necessarily global) solutions. The existence of local solutions in W1,∞(Ω) is assured under the assumption that initial data are non-negative functions in W01,∞(Ω), and that the mean curvature of the boundary ∂Ω of the domain Ω is non-positive. We here introduce a new method "L∞-energy method", which provides a main tool for our arguments and would be useful for other situations.

UR - http://www.scopus.com/inward/record.url?scp=0036014241&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036014241&partnerID=8YFLogxK

M3 - Article

VL - 8

SP - 647

EP - 670

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 3

ER -