In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces (-alpha,r)(q alpha) with (alpha, r) is an element of (1, 5/4) x [1,infinity] and q(alpha) = 3/alpha-1. Our work establishes a dichotomy of well-posedness and ill-posedness depending on r. Specifically, by combining the new endpoint bilinear estimates in (LxLT2)-L-q alpha, and L-T(infinity) (-alpha,1)(q alpha) and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in (-alpha,r)(q alpha) for r is an...
