Abstract :
Following our previous results on this subject [R.P. Agarwal, A. Prástaro, Geometry of PDE’s. III(I): Webs on PDE’s and
integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17 (2007) 239–266; R.P. Agarwal, A. Prástaro, Geometry of
PDE’s. III(II): Webs on PDE’s and integral bordism groups. Applications to Riemannian geometry PDE’s, Adv. Math. Sci. Appl.
17 (2007) 267–285; A. Prástaro, Geometry of PDE’s and Mechanics, World Scientific, Singapore, 1996; A. Prástaro, Quantum
and integral (co)bordism in partial differential equations, Acta Appl. Math. (5) (3) (1998) 243–302; A. Prástaro, (Co)bordism
groups in PDE’s, Acta Appl. Math. 59 (2) (1999) 111–201; A. Prástaro, Quantized Partial Differential Equations, World Scientific
Publishing Co, Singapore, 2004, 500 pp.; A. Prástaro, Geometry of PDE’s. I: Integral bordism groups in PDE’s, J. Math. Anal.
Appl. 319 (2006) 547–566; A. Prástaro, Geometry of PDE’s. II: Variational PDE’s and integral bordism groups, J. Math. Anal.
Appl. 321 (2006) 930–948; A. Prástaro, Th.M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal. Appl. 8 (2)
(2003) 259–278; I. Stakgold, Boundary Value Problems of Mathematical Physics, I, The MacMillan Company, New York, 1967;
I. Stakgold, Boundary Value Problems of Mathematical Physics, II, Collier–MacMillan, Canada, Ltd, Toronto, Ontario, 1968],
integral bordism groups of the Navier–Stokes equation are calculated for smooth, singular and weak solutions, respectively. Then
a characterization of global solutions is made on this ground. Enough conditions to assure existence of global smooth solutions are
given and related to nullity of integral characteristic numbers of the boundaries. Stability of global solutions are related to some
characteristic numbers of the space-like Cauchy data. Global solutions of variational problems constrained by (NS) are classified
by means of suitable integral bordism groups too.
© 2007 Elsevier Inc. All rights reserved.
Keywords :
Variational calculus constrained by the Navier , Navier–Stokes equation , Bordism groups , Global solutions , instability
