On the operator formulation of a class of problems of linear fluid dynamics
type: Item
instance of: scholarly article
title: On the operator formulation of a class of problems of linear fluid dynamics
zbMATH Open document ID: 0872.35089
DOI: 10.1007/BF02362671
author: M. Yu. Tsar'kov
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-06-26T00:00:00Z
review text: We propose a new operator formulation of a rather large class of problems of linear fluid dynamics. We give a result on the unique solvability of an evolution problem. In a bounded open set \(\Omega\) in \(\mathbb{R}^3\), we consider the equation \[ \rho u''_t- \rho(u_t'\times\omega)+\nabla p-(\nabla\rho\cdot u)\nabla\Pi= f(t,x),\tag{1} \] where \(u(t,x)\) and \(p(t,x)\) are unknown vector and scalar fields in \(\Omega\) and the functions \(\rho(x)\) and \(\Pi(x)\) and the vector field \(\omega(x)\) are defined in \(\Omega\) and have uniformly continuous first-order derivatives, the function \(\rho(x)\) being positive and bounded away from zero; \(f(t,x)\) is a given nonstationary vector field in \(\Omega\) used to state problems of linear fluid dynamics. For equation (1) we pose a boundary-value problem, state the natural concept of a generalized solution, and derive an operator equation equivalent to the generalized formulation of the problem. The basis of the method used here is the idea of the ``embedding'' of the boundary-value problem into an integral identity. The boundary condition by which the equation is supplemented contains two bilinear forms as parameters. By varying them, one can obtain as special cases a number of problems studied earlier. Moreover, the approach applied here leads rapidly to the goal and makes it possible to avoid excessive auxiliary constructions.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 46N20
zbMATH DE Number: 1026964
zbMATH Keywords: problems of linear fluid dynamics
zbMATH Keywords: unique solvability
zbMATH Keywords: operator equation
MaRDI profile type: Publication
cites work: Q4097611
cites work: Solvability in the small of nonstationary problems for incompressible ideal and viscous fluids and the case of vanishing viscosity
cites work: Q3993168
description: scientific article; zbMATH DE number 1026964