LIOUVILLE TYPE THEOREMS FOR THE STEADY AXIALLY SYMMETRIC NAVIER-STOKES AND MAGNETOHYDRODYNAMIC EQUATIONS

Abstract

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of [14] in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption vertical bar vertical bar u(r)/r 1({ur < 1/r})vertical bar vertical bar(L3/2(R3)) < C-# where C-# is a universal constant to be specified. In particular, if u(r)(r,z) >= -1/r for for all(r, z) is an element of [0, infinity) x R, then u equivalent to 0. Liouville theorems also hold if lim(vertical bar x vertical bar ->infinity) Gamma = 0 or Gamma is an element of L-q(R-3) for some q is an element of [2, infinity) where Gamma = ru(theta). We also established some interesting inequalities for Omega :- delta(z)u(r)-delta(r)u(z)/r, showing that del Omega can be bounded by Omega itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with u = u(r)(r, z)e(r) + u(theta)(r, z)e(theta) + u(z)(r, z)e(z),h - h(theta)(r, z)e(theta), indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure Phi = 1/2(vertical bar u vertical bar(2) + vertical bar h vertical bar(2)) + p for this special solution class.