Decay in Time of Incompressible Flows

In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stokes or MHD equations. We give a new proof for the time decay of the spatial $ L_2 $ norm of the solution, under the assumption that the solution of the heat equation with the same initial data decays. By first showing decay of the first derivatives of the solution, we avoid some technical difficulties of earlier proofs based on Fourier splitting.