Well-posedness for the dumbbell model of polymeric fluids
type: Item
instance of: scholarly article
title: Well-posedness for the dumbbell model of polymeric fluids
zbMATH Open document ID: 1060.35169
author: Pingwen Zhang
author: Weinan E
author: Tiejun Li
published in: Communications in Mathematical Physics
publication date: 2004-11-11T00:00:00Z
review text: Denote by \(u\) and \(p\) the velocity and pressure of the fluid, and \(Q\) the configuration of the spring, and hence the dumbbell, then \(Q\) obeys the following equation: \[ {\partial Q\over\partial t}+ (u\cdot\nabla) Q- (\nabla u)^TQ= -F(Q)+\dot w(t),\tag{1} \] where \(\dot w(t)\) is Gaussian white noise in time. The complete model for the full polymer-polymer system is then given by \[ {\partial u\over\partial t}+ (u\cdot\nabla)u+\nabla p= u+\nabla\tau,\quad \nabla u= 0,\tag{2} \] where \(\tau\) is the polymer contribution to stress, which is expressed via the Kramers expression \[ \tau(x,t)= E(F(Q)\otimes Q).\tag{3} \] The authors prove the following main result: Assume that the spring force \(F\) and the initial value satisfy the following conditions: (i) The function \(\gamma\) is \(C^\infty\) from \([0,+\infty)\) to \((0,+\infty)\), and \(\gamma'(| Q|^2)\geq 0\) and the derivative of \(F\) satisfies that \(|\nabla^m_{Q} F(Q)|\leq C(1+ | Q|^p)\) \((m= 0,1,2,3,4)\), where \(C\) is a constant and \(p\) is a certain nonnegative integer. (ii) \(\| u_0\|_{H^4}\leq \text{Const}.\), \(|\nabla^mQ_0|\leq \text{Const}.\) \((m= 0,1,2,3,4)\). Then there exists a \(T^*\) such that for \(t\leq T^*\), (1)--(3) has a unique strong solution \(u\) with \(u\in C^1([0, T^*], C^2(D))\cap L^2([0, T^*], H^5(D))\).
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76A05
Mathematics Subject Classification ID: 76M35
Mathematics Subject Classification ID: 82D60
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2114606
zbMATH Keywords: spring
zbMATH Keywords: dumbbell
zbMATH Keywords: polymer-polymer system
zbMATH Keywords: stress
zbMATH Keywords: unique strong solution
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
full work available at URL: s00220-004-1102-y
OpenAlex ID: W2132222929
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DOI: 10.1007/S00220-004-1102-Y
description: scientific article; zbMATH DE number 2114606