Functional Properties of the Navier-Stokes Operator and Bifurcation of Stationary Solutions: Planar Exterior Domains

The well-posedness of the stationary Navier-Stokes problem has been thoroughly investigated in the work of Leray [17], [18], Finn [8], Fujita [11], Ladyzhenskaya [16], Babenko [1] and many others. However, it is only fairly recently, with the input of Farwig [7], Galdi [12], Kobayashi and Shibata [14] and Shibata [25] that a functional setting has begun to emerge that seems suitable for the formulation of the steady-state Navier-Stokes problem on exterior domains. What we call here a suitable functional setting is a pair of Banach spaces X and Z and a mapping F: X → Z such that every solution of the equation F(x) = z, x ∈ X, z ∈ Z, yields a solution of the Navier-Stokes problem of interest. In this form, the problem may be given many unsubtle answers, but it becomes much trickier if, in addition, one insists on getting a mapping F having as many good properties as possible. Since mappings are primarily used to solve equations, a good property of a mapping F is anyone that opens up the possibility of taking advantage of known results to clarify the questions of existence, uniqueness (or nonuniqueness) or behavior of the solutions of the equation F(x) = z.