TY - JOUR

T1 - Hölder estimate for non-uniform parabolic equations in highly heterogeneous media

AU - Yeh, Li-Ming

PY - 2012/6

Y1 - 2012/6

N2 - Uniform bound for the solutions of non-uniform parabolic equations in highly heterogeneous media is concerned. The media considered are periodic and they consist of a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is associated with a positive number ∈, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the connected high permeability sub-region be of the order ∈ 2τ for τ ∈ (0,1]. It is proved that the Hölder norm of the solutions of the above non-uniform parabolic equations in the connected high permeability sub-region are bounded uniformly in . One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in ∈.

AB - Uniform bound for the solutions of non-uniform parabolic equations in highly heterogeneous media is concerned. The media considered are periodic and they consist of a connected high permeability sub-region and a disconnected matrix block subset with low permeability. Parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub-region and slow diffusion in the low permeability subset, and they form non-uniform parabolic equations. Each medium is associated with a positive number ∈, denoting the size ratio of matrix blocks to the whole domain of the medium. Let the permeability ratio of the matrix block subset to the connected high permeability sub-region be of the order ∈ 2τ for τ ∈ (0,1]. It is proved that the Hölder norm of the solutions of the above non-uniform parabolic equations in the connected high permeability sub-region are bounded uniformly in . One example also shows that the Hölder norm of the solutions in the disconnected subset may not be bounded uniformly in ∈.

KW - Highly heterogeneous media

KW - Infinitesimal generator

KW - Numerical range

KW - Paramatrix

KW - Pseudo-differential operator

KW - Strict solution

UR - http://www.scopus.com/inward/record.url?scp=84859108575&partnerID=8YFLogxK

U2 - 10.1016/j.na.2012.01.027

DO - 10.1016/j.na.2012.01.027

M3 - Article

AN - SCOPUS:84859108575

SN - 0362-546X

VL - 75

SP - 3723

EP - 3745

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 9

ER -