A survey of analytical studies of steady viscous incompressible flows (2000–2004)

General studies of the Navier–Stokes equations and their analytical solutions describing new vortex structures in viscous flows are surveyed. In the case of plane-parallel steady flows, homogeneous forms of the Navier–Stokes equations are derived, the properties of boundary value problems are examined, the domain of analytical solutions to Cauchy problems is estimated, solutions to Cauchy problems that describe separated flows are found, and the interaction between vortex systems and channel flows is analyzed in the Stokes approximation. In the axisymmetric case, the Kovalevskaya theorem is generalized to Cauchy problems with data lying on the axis of symmetry, which is a singular line for the Navier–Stokes equations; the domain of analytical solutions for such a problem is estimated; in the special case, a class of exact solutions is obtained that describe, in particular, systems of axial vortex structures such as Hill spherical vortices; an Cauchy problem is solved analytically in the general case; and a flow past a sphere with the no-slip condition set on the sphere is considered.