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The blow‐up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow
Author(s) -
Yuan Baoquan,
Li Rui
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3352
Subject(s) - inviscid flow , mathematics , ideal (ethics) , compressibility , viscoelasticity , singularity , flow (mathematics) , mathematical analysis , incompressible flow , dissipation , laplace operator , space (punctuation) , mathematical physics , classical mechanics , geometry , mechanics , physics , thermodynamics , law , philosophy , linguistics , political science
In this paper, we prove two blow‐up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (−Δ) α u in the spaceR n , n = 2 , 3 and one for the inviscid Oldroyd model. Assume that ( u ( t , x ), F ( t , x )) is a smooth solution to the generalized Oldroyd model in [0, T ); then, the solution ( u ( t , x ), F ( t , x )) does not develop singularity until t = T provided ∇ u ∈ L 10 , T ;B ̇∞ , ∞ 0. For the ideal impressible viscoelastic flow, it is shown that the smooth solution ( u , F ) can be extended beyond T if ( ∇ u , ∇ F ) ∈ L 10 , T ;B ̇∞ , ∞ 0, which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.