L2 DECAY OF WEAK SOLUTIONS FOR THE NAVIER-STOKES EQUATIONS IN GENERAL DOMAINS

Abstract

Let u be a weak solution of the in-stationary Navier-Stokes equations in a completely general domain in R3. Firstly, we prove that the time decay rates of the weak solution u in the L2-norm like ones of the solutions for the homogeneous Stokes system taking the same initial value in which the decay exponent is less than 34 . Secondly, we show that under some additive conditions on the initial value, then u coincides with the solution of the homogeneous Stokes system when time tends to infinity. Our proofs use the theory about the uniqueness arguments and time decay rates of strong solutions for the Navier-Stokes equations in the general domain when the initial value is small enough.