THE NAVIER-STOKES EQUATIONS AND WEAK HERZ SPACES

In this paper, we discuss the Cauchy problem for Navier-Stokes equations in homogeneous weak Herz spaces W(K) over dot(p,q)(alpha)(R-n). More precisely, we construct the solution in the class L-infinity(0,T; W(K) over dot(p,q)(alpha)) with the initial data in W(K) over dot(p,q)(alpha). Further, we consider the blow-up phenomena of time-local solutions and the uniqueness of global solutions with large initial data in W(K) over dot(p,q)(alpha). Also, we give several embeddings of weak Herz spaces into homogeneous Besov spaces (B) over dot(p,infinity)(-alpha)(R-n) (alpha > 0), or bmo(-1) (R-n).