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

This paper studies the spatially periodic Navier-Stokes flows in R-2 driven by a unidirectional external force. This dynamical system admits a steady-state solution u(0) for all Reynolds numbers. u(0) is the basic flow in our consideration, and primary bifurcations of u(0) are investigated. In particular, it is found that there exists a flow invariant subspace containing cos(mx + ny) or sin(mx + ny), and the occurrence of stability and bifurcations of u(0) in such a subspace essentially depends on the choice of the integers m and n. Our findings are obtained by analysis together with numerical computation.