In this paper we prove some properties of the maximal solution of Navier-Stokes equations. If the maximum time T * ν is finite, we establish that the growth of u(t) Ḣs is at least of the order of (T * ν − t) −s/3 (see Eq.(1.4)), also we give some new blow-up results. Specific properties and standard techniques are used.

In this paper we prove, if u ∈ C([0, ∞), X −1 (R 3 )) is global solution of 3D Navier-Stokes equations, then u(t) X −1 decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

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In [5], Benameur proved a blow-up result of the non regular solution of (N SE) in the Sobolev-Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev-Gevrey spaces with less regularity on the initial condition. Fourier analysis is used.

In this paper, we study the sub-critical dissipative quasi-geostrophic equations (S α ). We prove that there exists a unique local-in-time solution for any large initial data θ 0 in the space X 1−2α (R 2 ) defined by (1). Moreover, we show that (S α ) has a global solution in time if the norms of the initial data in X 1−2α (R 2 ) are bounded by 1/4. Also, we prove a blow-up criterion of the local-in-time solution of (S α ).