On the impossibility, in some cases, of the leray-hopf condition for energy estimates

Abstract

Current proofs of time independent energy bounds for solutions of the time dependent Navier-Stokes equations, and of bounds for the Dirichlet norms of steady solutions, are dependent upon the construction of an extension of the prescribed boundary values into the domain that satisfies the inequality (1.1) below, for a value of ? less than the kinematic viscosity. It is known from the papers of Leray (J Math Pure Appl 12:1-82, 1993), Hopf (Math Ann 117:764-775, 1941) and Finn (Acta Math 105:197-244, 1961) that such a construction is always possible if the net flux of the boundary values across each individual component of the boundary is zero. On the other hand, the nonexistence of such an extension, for small values of ?, has been shown by Takeshita (Pac J Math 157:151-158, 1993) for any two or three-dimensional annular domain, when the boundary values have a net inflow toward the origin across each component of the boundary. Here, we prove a similar result for boundary values that have a net outflow away from the origin across each component of the boundary. The proof utilizes a class of test functions that can detect and measure deformation. It appears likely that much of our reasoning can be applied to other multiply connected domains. © 2010 Birkhäuser Verlag Basel/Switzerland.