The Method of Partial Regularity as Applied to the Navier-Stokes Equations

Abstract

The solutions of a system of partial differential equations are frequently studied in two steps: first one proves the existence of a weak solution in a suitable Sobolev space; then one proves the regularity of this weak solution. For nonlinear systems the second step may be too difficult--indeed. it may be false. the solutions may have singularities. In such cases one seeks a partial regularity theorem. restricting the size of the set of possible singularities. and one attempts the local description. to leading order. of the behavior near a singularity. This approach has been used with great success in the theory of codimension one area-minimizing surfaces. developed by de Giorgi. Almgren. and others. There the weak solutions are described using integral currents or sets of finite perimeter; the singular set has codimension seven; and near any singular point the solution behaves locally like a minimal cone. The literature on this and other geometric applications is extensive; a rather accessible introduction is given in [3]. A similar approach has been used to study elliptic equations and systems. One recent success is the application by Schoen and Uhlenbeck to harmonic mappings [13]; other aspects of recent work are reviewed by Giaquinta and Giusti in [1].