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Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations. (English) Zbl 1078.35087

The authors consider the Navier-Stokes equations in n , i.e.

u t -Δu+(u·)u+p=0in n ×(0,T),divu=0in n ×(0,T),u(0)=ain n ·

They prove that a strong solution which satisfies

0 T u(t) F ˙ , -α 2 1-α dt<forsome0<α<1(1)

can be continued beyond T. Here, F ˙ p,q s is a homogeneous Triebel-Lizorkin space. Furthermore, they show that a weak solution satisfying (1) is strong in (ε,T) provided that n4. The main tool in proving the above results is a Hölder type inequality in Triebel-Lizorkin spaces.


MSC:
35Q30Stokes and Navier-Stokes equations
42B25Maximal functions, Littlewood-Paley theory
76D05Navier-Stokes equations (fluid dynamics)