Bifurcations of a steady-state solution to the two-dimensional Navier-Stokes equations (Q1300619)

A plane spatially periodic Navier-Stokes flow is considered, produced by a special one-dimensional exterior force. The system admits a steady-state solution. All primary bifurcations of the above system are studied, by using the flow invariant subspaces. A formulation in vorticity is used, and the rotation of the exterior forces is decomposed into a Fourier series to obtain the corresponding development for the flow. Thus the initial value system becomes an infinite-dimensional dynamical system for the considered ``basis''. A specific subspace is studied corresponding to the ``basis'' \(\cos(mx+ ny)\) or \(\sin(mx+ ny)\). The bifurcation of the above steady state is studied, as function of \(m\) and \(n\). In the last part, a numerical computation is also used to characterize the invariant subspaces.