About lifespan of regular solutions of equations related to viscoelastic fluids
type: Item
instance of: scholarly article
zbMATH Open document ID: 1007.76003
DOI: 10.1137/S0036141099359317
author: Jean-Yves Chemin
author: Nader Masmoudi
published in: SIAM Journal on Mathematical Analysis
publication date: 2001-06-21T00:00:00Z
Mathematics Subject Classification ID: 76A10
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 1608847
zbMATH Keywords: Oldroyd model
zbMATH Keywords: well-posedness
zbMATH Keywords: Besov spaces
zbMATH Keywords: viscoelastic fluids
zbMATH Keywords: stress tensor
zbMATH Keywords: Cauchy problem
zbMATH Keywords: periodic problem
zbMATH Keywords: unique solvability
zbMATH Keywords: maximal time interval
zbMATH Keywords: necessary blow-up condition
zbMATH Keywords: global solvability
MaRDI profile type: Publication
full work available at URL: s0036141099359317
OpenAlex ID: W1973779621
title: About lifespan of regular solutions of equations related to viscoelastic fluids
review text: The authors consider the system of equations \( v_t +v\cdot\nabla v-\nu\Delta v=\mu _1 \nabla\cdot\tau,\) \(\tau_t +v\cdot\nabla\tau +a\tau +Q(\tau,\nabla v)=\mu _2 D(v),\) \(\text{div }v=0\) describing flows of viscoelastic fluids. Here \(\tau\) is the elastic part of stress tensor, and \(Q(\tau,\nabla v)=\tau W(v)-W(v)\tau -b[D(v)\tau +\tau D(v)],\) \(2D(v)_{ij}=\partial _j v_i +\partial _i v_j ,\) \(2W(v)_{ij}=\partial _j v_i -\partial _i v_j .\) The constants \(\nu\), \(\mu _i\), \(a\) and \(b\) are assumed to be non-negative. The authors discuss Cauchy problem and \(x\)-periodic problem. A unique solvability theorem is proved for some maximal time interval \((0,T)\) under the necessary blow-up condition \(T<\infty \Longrightarrow\int\limits_0 ^T \frac{\mu_1 }{\nu\mu_2 }\|\tau\|_{\infty}^2 +\|\nabla v\|_{\infty}dt=\infty.\) In two space dimensions, the necessary blow-up condition writes \(T<\infty \Longrightarrow\int\limits_0 ^T \|\tau\|_{\infty} +|b|\|\tau\|_{2}^2 dt=\infty.\) Finally, with the use of Besov spaces, the authors prove a unique global solvability for small initial data.
reviewed by: V. V. Shelukhin
description: scientific article; zbMATH DE number 1608847