Decay bounds for magnetohydrodynamic geophysical flow

This paper establishes exponential decay bounds for steady magnetohydrodynamic geophysical finite pipe flow when homogeneous lateral surface boundary conditions are applied. In the spirit of earlier work of C.O. Horgan, L.T. Wheeler [Spatial decay estimates for the Navier-Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math. 35 (1978) 97-116] and K.A. Ames, L.E. Payne [Decay estimates in steady pipe flow, SIAM J. Math. Anal. 20 (1989) 789-815] the decay to fully developed flow as a function of distance from the entry section is investigated. Here, it is not assumed that the flow is symmetric like in the work of J.C. Song, [Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Anal. 54 (2003) 1029-1044] and it is fully developed at the exit section. Using a coupling parameter, we derive an integro-differential inequality that leads to estimates for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and the fully developed flow in a portion of the pipe near the exit section. This paper also indicates how to bound the total energy.