A periodic boundary value problem for the equation of transonic gas dynamics
type: Item
instance of: scholarly article
title: A periodic boundary value problem for the equation of transonic gas dynamics
zbMATH Open document ID: 0917.35098
DOI: 10.1007/BF02674289
author: S. N. Glazatov
published in: Siberian Mathematical Journal
publication date: 1999-05-10T00:00:00Z
review text: The author poses the following boundary value problem in the cylindrical domain \(Q = D \times (0,L)\), \(D \subset \mathbb{R}^2\) (\(D\) is a bounded domain of the \((y_1,y_2)\)-plane with smooth boundary): Find a solution to the equation \[ u_xu_{xx}-u_{y_1y_1}-u_{y_2y_2}+ au_x=f(x,y)\quad (a>0) \tag{1} \] satisfying the conditions \[ u|{}_{x=0} = u|{}_{x=L}, \quad u_x|{}_{x=0} = u_x|{}_{x=L}, \quad \frac {\partial u}{\partial \vec{n}}\biggr|{}_{\partial D \times (0,L)} = 0, \] where \(\vec{n}\) is the normal to \(\partial D\). Under the conditions \[ D^j_xf\big|{}_{x=0}=D^j_xf\big|{}_{x=L},\quad j=0,1,2, \quad \int_Qf dxdy=0, \] \[ |{}|{}|{}f|{}|{}|{}\equiv \biggl( \sum^{3}_{j=0} \bigl\|{}D^j_x f\bigr\|{}^2_{L_2(Q)} +\|{}\nabla_y f_x\|{}^2_{L_2(Q)}\biggl)^{1\over 2}<\alpha, \] where \(\alpha>0\) is some positive number depending on \(a\) and the cylinder \(Q\), the problem is proven to have a regular solution. Moreover, this solution is unique, provided that some additional conditions are satisfied. A characteristic feature of the problem is its ``indifference'' to the type of equation \((1)\), i.e., to the sign of the function \(u_x\).
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76H05
Mathematics Subject Classification ID: 35M10
zbMATH DE Number: 1285614
zbMATH Keywords: nonlinear variable-type equation
zbMATH Keywords: periodic boundary value problem
zbMATH Keywords: smallness conditions
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02674289
OpenAlex ID: W2087017457
description: scientific article; zbMATH DE number 1285614