The Hopf bifurcation with symmetry for the Navier-Stokes equations in \((L_ p(\Omega))^ n\), with application to plane Poiseuille flow
type: Item
instance of: scholarly article
title: The Hopf bifurcation with symmetry for the Navier-Stokes equations in \((L_ p(\Omega))^ n\), with application to plane Poiseuille flow
zbMATH Open document ID: 0686.35010
DOI: 10.1007/BF00281352
author: Thomas J. Bridges
published in: Archive for Rational Mechanics and Analysis
publication date: 1989-01-01T00:00:00Z
review text: Let \(\Omega \subset {\mathbb{R}}^ n\), \(n=2,3\), be such that each direction is either bounded or infinite and periodic, \(k=(k_ 1,...,k_ n)\) a wave number vector, U a solution of the stationary Navier-Stokes equations. The governing equations are \[ (*)\quad \sum^{n}_{i=1}k_ i\partial u_ i/\partial x_ i=0, \] \[ \partial u_ i/\partial t+\sum^{n}_{j=1}k_ j\partial /\partial x_ j(U_ iu_ j+u_ iU_ j+u_ iu_ j)+k_ i\partial p/\partial x_ i-\lambda \Delta u_ i=0,\quad i=1,...,n. \] Let \(C^{\infty}_{0,\sigma}(\Omega):=\{u\in C^{\infty}_ 0(\Omega):\) \(\sum^{n}_{i=1}k_ i\partial u_ i/\partial x_ i=0\}\), \(X_ p:=closure\) of \(C^{\infty}_{0,\sigma}(\Omega)\) in \((L_ p(\Omega))^ n\). (*) can be formulated as abstract equation in \(X_ p:\) \[ (**)\quad du/dt+Au=M(U)u+N(u,u),\quad t>0. \] Using the fact that the Stokes operator A generates an analytic semigroup in X for \(1<p<\infty\) the author proves that (**) has a unique local solution \(u\in C((0,\tau),D(A))\cap C((0,\tau),X_ p)\) for \(p>n\) satisfying \(u(0)=u_ 0\) for any \(u_ 0\in X_ p\). Let Z be the complexification of \(D(A^{1/2})\) in \(X_ p\), let \(C_{2\pi}(R,Z)\) be the space of continuous \(2\pi\)-periodic functions mapping R into Z. To prove the occurrence of Hopf bifurcation to (**) with \(\lambda\) as bifurcation parameter the Lyapunov-Schmidt method is applied to some integral equation which is generated by a compact operator. This leads to a finite-dimensional branching equation which is equivariant with respect to some spatial symmetry group. In the last section the theory is applied to the problem of time periodic solution bifurcating from planar Poiseuille flow in the presence of spatial symmetry \(SO(2)\times O(2)\times S\) where O(2) invariance is in the spanwise direction. In this case, the author proves that generically there is a bifurcation to oblique travelling waves as well as standing-travelling waves (stationary in the spanwise direction, travelling in the down-stream direction). There a…
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 4124183
zbMATH Keywords: Poiseuille flow
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: symmetry
zbMATH Keywords: analytic semigroup
zbMATH Keywords: Lyapunov-Schmidt method
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124183