Free boundary problem in a polymer solution model
type: Item
instance of: scholarly article
title: Free boundary problem in a polymer solution model
zbMATH Open document ID: 1472.35423
DOI: 10.1134/S1061920821010106
author: Anna G. Petrova
author: Vladislav V. Pukhnachev
published in: Russian Journal of Mathematical Physics
publication date: 2021-05-05T00:00:00Z
review text: The authors consider the integro-differential equation \[ \frac{\partial w}{\partial t}+\frac{\partial w}{\partial y}\int_{0}^{y}w(z,t)dz-w^{2}=\frac{\partial^{2}w}{\partial y^{2}}+\gamma (\frac{\partial^{3}w}{\partial y^{2}\partial t}+\frac{\partial^{3}w}{\partial y^{3}}\int_{0}^{y}w(z,t)dz- \frac{\partial^{2}w}{\partial y^{2}}), \] posed in the domain \(\Omega_{T} = \{y,t:0 < y < h(t)\), \(0\leq t\leq T\}\). This model accounts for the flow of a mixture of water and polymer. This equation is completed with: \(\frac{dh}{dt}=\int_{0}^{h}w(y,t)dy\). The boundary conditions \(w(0,t)=0\) and \(\frac{\partial w}{\partial y}+\gamma (\frac{\partial^{2}w}{\partial y\partial t}+ \frac{\partial^{2}w}{\partial y^{2}}\int_{0}^{h}w(y,t)dy-w\frac{\partial w}{\partial y})(h(t),t)=0\) are added, together with the initial conditions \( w(y,0)=w_{0}(y)\), \(0\leq y\leq 1\), \(h(0)=1\). Here \(\gamma >0\) is a constant and \(w_{0}\) is a smooth (\(C^{3}\)) function of \(y\) satisfying the conditions \( w_{0}(0)=w_{0}^{\prime}(1)=0\). The first main result proves the existence of a local in time strong solution (\(h\in C^{1}([0,t^{\ast}])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,t^{\ast}])\)) to this problem. If the initial condition further satisfies \(w_{0}(y)\leq 0\), \(w_{0}(y)-\gamma w_{0}^{\prime \prime}(y)\leq 0\), the authors prove the existence of a classical solution \(h\in C^{1}([0,T])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,T])\) to the above problem. Both existence results are obtained through appropriate transformations and using Schauder's theorem. The authors then consider the case where \(\gamma\) tends to 0 and they observe that the problem turns into that of the deformation of a strip of viscous fluid. They here prove that the solution to this problem is destructed in finite time. They finally introduce asymptotic expansions with respect to \(\gamma\) and they express the second term of this asymptotic expansion.
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 82D60
Mathematics Subject Classification ID: 76A20
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 7344450
zbMATH Keywords: integro-differential equation
zbMATH Keywords: dissolved polymer
zbMATH Keywords: existence result
reviewed by: Alain Brillard
MaRDI profile type: Publication
cites work: Mathematical models of convection
cites work: Q3945260
cites work: On the unique solvability of the problem of the flow of an aqueous solution of polymers near a critical point
cites work: Q4045187
full work available at URL: s1061920821010106
OpenAlex ID: W3137942372
description: scientific article; zbMATH DE number 7344450