A transformation of coordinates in a problem with a free boundary
type: Item
instance of: scholarly article
title: A transformation of coordinates in a problem with a free boundary
zbMATH Open document ID: 0766.35084
DOI: 10.1007/BF00970473
published in: Siberian Mathematical Journal
publication date: 1992-06-26T00:00:00Z
review text: The paper deals with the nonstationary flow of a compressible fluid with power dependency of the tangential tension on the sliding velocity under resilient filtering conditions. At the initial moment the fluid covers the half-space \(x<0\), while in the motion it is bounded by the solid wall \(x=0\). The case is studied when the fluid density decreases interior to the half-space, i.e. the equation \[ u_ t={\partial \over {\partial x}} u_ x^ m, \qquad x<0, \qquad 0<t\leq T \] with initial boundary conditions \(u(x,0)=u_ 0(x)\), \(u_ 0(x)>0\), \(x\in(-1,0]\), \(u_ 0(x)\equiv 0\), \(x\leq-1\), \(u(0,t)=p\), \(t\geq 0\). The problem is solved by means of the transformation of coordinates \((x=X(\xi,t))\) connected with equipotentials, i.e. the evolution of a fluid particle is defined as its motion along the path of the constant density value \(u(X(\xi,t),t)=u_ 0(\xi)\). After the transformation a nonclassical boundary-value problem is obtained for the investigation of which the author uses the theory of partial differential equations: he defines the notion of generalized solution of this boundary-value problem, proves its existence, obtains estimates. Using the obtained results he proves the existence of a solution to the initial problem, its continuity and positiveness.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 20553
zbMATH Keywords: nonstationary flow
zbMATH Keywords: compressible fluid
zbMATH Keywords: generalized solution
zbMATH Keywords: existence
zbMATH Keywords: continuity
zbMATH Keywords: positiveness
author: Vladimir N. Grebenev
reviewed by: Bronius Kvedaras
MaRDI profile type: Publication
cites work: Q5812835
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full work available at URL: https://doi.org/10.1007/bf00970473
OpenAlex ID: W2016973633
description: scientific article; zbMATH DE number 20553