The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions

Yemin Chen; Ping Zhang

doi:10.1080/03605300600858960

The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions

Yemin Chen, Ping Zhang^*

^*Corresponding author for this work

School of Mathematics and Statistics

CAS - Academy of Mathematics and System Sciences

Research output: Contribution to journal › Article › peer-review

102 Citations (Scopus)

Abstract

In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this system blows up at some finite time T*, then it is necessary that ∫₀^T* ∥∇υ(t)∥_L∞ dt = ∞. Furthermore, we prove the global existence of smooth solutions to this system provided that initial data is sufficiently close to the equilibrium state.

Original language	English
Pages (from-to)	1793-1810
Number of pages	18
Journal	Communications in Partial Differential Equations
Volume	31
Issue number	12
DOIs	https://doi.org/10.1080/03605300600858960
Publication status	Published - 1 Dec 2006

Keywords

Blow up principle
Global small solutions
Oldroyd model

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10.1080/03605300600858960

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abstract = "In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this system blows up at some finite time T*, then it is necessary that ∫0T* ∥∇υ(t)∥L∞ dt = ∞. Furthermore, we prove the global existence of smooth solutions to this system provided that initial data is sufficiently close to the equilibrium state.",

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N2 - In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this system blows up at some finite time T*, then it is necessary that ∫0T* ∥∇υ(t)∥L∞ dt = ∞. Furthermore, we prove the global existence of smooth solutions to this system provided that initial data is sufficiently close to the equilibrium state.

AB - In this article, we study the Cauchy problem of the viscoelastic fluid system of Oldroyd model in space dimension greater than 1. In particular, we prove that if the smooth solution (U,v) to this system blows up at some finite time T*, then it is necessary that ∫0T* ∥∇υ(t)∥L∞ dt = ∞. Furthermore, we prove the global existence of smooth solutions to this system provided that initial data is sufficiently close to the equilibrium state.

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The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions

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Keywords

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