A Uniqueness Theorem for the Unbounded Classical Solution of the Nonstationary Navier-Stokes Equations in R3

Some new results on the nonstationary Navier-Stokes equations are presented. Our results connect the well known, functional analytic theory for the Navier-Stokes equations with the blow-up solutions which were newly found by Ohkitani and others [S. Childress et al., J. Fluid Mech. 203 (1989), 1-22; K. Ohkitani, J. Phys. Soc. Japan59 (1990), 3811-3814; J. T. Stuart, in "Symposium to Honor C. C. Lin (D. J. Benny, F. H. Shu, and C. Yuan, Eds.), pp. 81-95, World Scientific, 1987]. Actually we consider unbounded solutions and prove a generalization of Graffi′s uniqueness theorem [Ann. Mat. Pura Appl (4) 50 (1960), 379-387].