On periodic transonic flows of a viscous gas
type: Item
instance of: scholarly article
title: On periodic transonic flows of a viscous gas
zbMATH Open document ID: 0868.35088
DOI: 10.1007/BF02674900
author: S. N. Glazatov
published in: Siberian Mathematical Journal
publication date: 1997-07-22T00:00:00Z
review text: The equation considered in the paper under review was proposed by \textit{O. S. Ryzhov} [Prikl. Mat. Mekh. 29, 1004-1014 (1965; Zbl 0173.28104)]\ for describing transonic flow of a viscous gas: \[ Au\equiv-\mu u_{xxx}+u_xu_{xx}-\Delta_yu=f(x,y), \] where the parameter \(\mu>0\) characterizes viscosity and thermal conductivity of the gas. The author studies the problem of finding a solution to the equation in the cylinder \(Q=(0,L)\times D\), subject to the conditions \[ D_x^j|_{x=0}=D_x^j|_{x=L},\quad j=0,1,2,\quad{\partial u\over\partial n}|_{(0,L)\times\gamma}=0, \] where \(D\subset \mathbb{R}^2\) is a bounded domain with \(C^2\)-smooth boundary \(\gamma\). It is proved that there exists a solution, provided that \(|f|_{L_2(Q)}+|f_x|_{L_2(Q)}<\alpha\), where \(\alpha\) is some positive number depending on \(\mu\) and \(Q\). It is shown that the solution is unique in some function class. The proofs are accomplished by considering a linearized problem and fixed point arguments.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35G15
Mathematics Subject Classification ID: 76H05
zbMATH DE Number: 1037962
zbMATH Keywords: viscous transonic flow
zbMATH Keywords: nonlinear third-order partial differential equation
zbMATH Keywords: periodic solutions
MaRDI profile type: Publication
cites work: Q5560985
cites work: Q4144365
full work available at URL: https://doi.org/10.1007/bf02674900
OpenAlex ID: W2094562659
description: scientific article; zbMATH DE number 1037962