Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

Abstract

The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all t > 0 step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the L p decay estimate for the solution u(•, t) and all its derivatives for generalized Navier-Stokes equations with max{1, 3+q 6 } < α ≤ 1 2 + min{ 3 q − 3 p , 3 2p }.