REGULARITY CRITERIA FOR THE GENERALIZED NAVIER-STOKES AND RELATED EQUATIONS

We use the maximum principle type estimate and interpolation inequality on Besov spaces to show some regularity criteria for the generalized Navier-Stokes equations, the quasi-geostrophic equations, and the harmonic heat flow.


Introduction
In this paper, we study the regularity criteria for the generalized Navier-Stokes equations, quasi-geostrophic equations, and harmonic heat flow equations.
To prove this theorem, we will use the following maximum principle type estimates: Lemma 1.1. Let 0 < < 5/2, 2 ≤ p < ∞ and θ be a smooth function on R n . We have: Next, we study the following 2-D quasi-geostrophic equations [9,7,14,18]: where θ is a scalar function representing temperature, u is the velocity field of the fluid. The Riesz transforms R 1 and R 2 are defined by The case 1/2 < ≤ 1, is called subcritical since smooth solutions are known to exist globally in time [8]. The case = 1 2 is called critical since there is a balance between the dissipation and the non-linear term, therefore is a good model for the 3D Navier-Stokes equations. Very recently, Kiselev-Nazarov-Volberg [15] showed the existence of global smooth solutions. The case 0 < < 1 2 is called supercritical and is harder to deal with compared to the other cases. In order to make the similarities to the 3D Navier-Stokes equations more apparent we apply the operation Then, we observe that ∇ ⊥ θ has the role of vorticity. Recently, D. Chae [4] gives the regularity condition: J. Yuan [26] improves it to the following condition: In this paper, we will prove: Then there is no singularity up to T .
Finally, we consider the regularity problem for smooth solutions to the time-dependent harmonic heat flow from R n into a unit sphere S m : (1.14) The regularity of the weak solution fails in general because of the existence of a blow-up solution for large initial data. The example for the map from B 1 (0) ⊂ R n to a sphere was shown by Coron-Ghidaglia [10] for n ≥ 3 and Chang-Ding-Ye [6] for n = 2. However, some smallness assumption on the initial data or integrability condition on the solution itself are sufficient to give the regularity. Ogawa [22] showed the following regularity conditions: Here,Ḟ 0 ∞,2 is the homogeneous Triebel-Lizorkin space. We will improve (1.15) and (1.16) to the following results.  (1.14). Assume that one of the following conditions is satisfied: (1.17) (1.18) Then there is no singularity up to T .
, our result improves that of Ogawa [22] when 2 ≤ n ≤ 4. Moreover, our proof of (1.19) below is simple. Theorem 1.1 is proved in Section 2, Theorem 1.2 is proved in Section 3, and the final Section 4 is devoted to the proof of Theorem 1.3.

Proof of Theorem 1.3
This section is devoted to the proof of Theorem 1.3.
(2) We assume that the condition (1.18) holds true. Applying ∆ to equation (1.14) 1 , we find that Multiplying the above equation by |∆u| p−2 ∆u, integrating by parts, and using Hölder's inequality, we see that for any > 0 by Young's inequality.
Here we have used the Gagliardo-Nirenberg inequality: We use (2.3) for α = 1 to estimate with r satisfying 2 r + n p = 2. Inserting (4.6) into (4.4) and taking small enough, we get which implies ∆u L ∞ (0,T ;L p ) ≤ C and thus the solution u is regular.
This completes the proof.