Bifurcations of a Steady-State Solution to the Two-Dimensional Navier–Stokes Equations

This paper studies the spatially periodic Navier–Stokes flows in R2 driven by a unidirectional external force. This dynamical system admits a steady-state solution u0 for all Reynolds numbers. u0 is the basic flow in our consideration, and primary bifurcations of u0 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 u0 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.