Studies on transonic flow problems by nonlinear variational inequalities
type: Item
instance of: scholarly article
title: Studies on transonic flow problems by nonlinear variational inequalities
zbMATH Open document ID: 0655.76050
DOI: 10.4171/ZAA/263
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1987-01-01T00:00:00Z
review text: The author considers an irrotational, steady and isentropic flow of a non-viscous, compressible fluid in a bounded, simply connected domain \(\Omega \subset R^ N\) (N\(\geq 2)\), governed by the standard partial differential equation for the velocity potential u, to which a suitable entropy condition, a bound for the gas velocity, and the boundary conditions for u, must be added. Weak formulations of the boundary value problems having physical interest are given. There is to find an \(u\in V\) such that: \[ \int_{\Omega}\rho (| \nabla u|^ 2)\nabla u\nabla v dx=\int_{R}gv do\quad for\quad all\quad v\in V, \] where V has different expressions according to the particular considered problem. The above equation is the Euler- Lagrange-equation for the variational problem: \[ F(v):=\int_{\Omega}(\int^{| \nabla v|^ 2}_{0}\rho (q)dq)dx-\int_{R}gv do\to_{v\in V}. \] Now F(v) must be minimized over all \(v\in V\), with additional constraints, that means: F(v)\(\to_{v\in K}\), where K is a well defined subset of V. If \(u\in K\) is a minimum point of this variational problem, then u satisfies for all \(v\in K\) the variational inequality: \[ DF(u,u-v)=\int_{\Omega}\rho (| \nabla u|^ 2)\nabla u\nabla (u-v)dx-\int_{R}g(u-v)do\leq 0. \] Using Katchanov's method and the compactness of the convex set this variational inequality is solved. Furthermore, a result on flows in the subsonic region is given.
Mathematics Subject Classification ID: 76H05
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 4070525
zbMATH Keywords: non-viscous, compressible fluid
zbMATH Keywords: bounded, simply connected domain
zbMATH Keywords: boundary value problems
zbMATH Keywords: Euler-Lagrange-equation
zbMATH Keywords: variational inequality
author: Hans-Peter Gittel
reviewed by: Silvio Nocilla
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4070525