Pointwise Asymptotic Stability of Steady Fluid Motions

We study pointwise asymptotic stability of steady incompressible viscous fluids. The region of the motion is bounded. Our results of stability are based on the maximum modulus theorem that we prove for solutions of the Navier–Stokes equations. The asymptotic stability is based on a variational formulation. Since the region of the motion is bounded, the time decay is of exponential type. Of course suitable assumptions are made about the smallness of the size of the uniform norm of the perturbations at the initial data. With no restrictions, we are able only to prove an existence theorem of the perturbation local in time.