The linearized Crocco equation
type: Item
instance of: scholarly article
title: The linearized Crocco equation
zbMATH Open document ID: 1214.35035
author: J-P Raymond
author: Jean-Marie Buchot
published in: Journal of Mathematical Fluid Mechanics
publication date: 2007-02-14T00:00:00Z
review text: The authors study a two-dimensional boundary layer over a flat plate in the context of the linearized Crocco equation around the steady state solution. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations. The main aim of this paper is to prove the existence and uniqueness of a weak solutions associated with the adjoint linearized Crocco equation with the homogeneous boundary conditions. This equation is doubly degenerate: the coefficient in front of \(\partial_x\) vanishes on a part of the boundary, and the coefficient of the elliptic operatore vanishes in another part of the boundary, \(x\)-variable plays the role of time variable. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operatore is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument. This is also used to prove a priori estimates, which cannot be obtained in a classical way. Finally, the results obtained are compared with other existing in the literature.
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D10
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 5126601
zbMATH Keywords: boundary layer equations
zbMATH Keywords: Prandtl equations
zbMATH Keywords: homogeneous boundary conditions
MaRDI profile type: Publication
full work available at URL: s00021-005-0186-2
OpenAlex ID: W2016128309
DOI: 10.1007/S00021-005-0186-2
description: scientific article; zbMATH DE number 5126601