The Hopf bifurcation with symmetry for the Navier-Stokes equations in \((L_ p(\Omega))^ n\), with application to plane Poiseuille flow
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 are points of degeneracy on the neutral surface. These points are determined numerically. An analysis in the neighbourhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.
scientific article; zbMATH DE number 4124183
scientific article