Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field

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Abstract

We establish some regularity criteria for the solutions to the Navier–Stokes equations in the full three-dimensional space in terms of one directional derivative of the velocity field. Revising the method used by Zujin Zhang (2018), we show that a weak solution u is regular on (0, T] provided that ux3Lp(0,T;Lq(R3)) with s=2 for 3q6, 116<s2 for 6q66s11 where s=2p+3q. They improve the known results 2p+3q=32 for 2q, 2p+3q85+911q for 52q< and 2p+3q1411+35q for 4q< .

Section snippets

Introduction and main results

We consider sufficient conditions for the regularity of solutions of the Cauchy problem for the Navier–Stokes equations in R3 utΔu+uu+p=0 in R3×(0,),u=0 in R3×(0,),ut=0=u0, where u=u1,u2,u3:R3×(0,T)R3 is the velocity field, p:R3×(0,T)R is a scalar pressure and u0 is the initial velocity field. We recall some well-known function spaces, the definitions of weak and strong solutions to (1.1) and introduce some notations before describing the main results. Throughout the paper, we

The proof of the main results

The proofs of Theorem 1.1, Theorem 1.3 are based on the method used in [10]. We define the quantities L and J (see [5], [8]) and then prove that L and J are uniformly bounded in time. Let T=sup{τ>0;u is regular on (0,τ)}. Since u0H1,u is regular on some positive time interval and T is either equal to infinity (in which case the proof is finished) or it is a positive number and u is regular on 0,T, that is uLloc0,T;L2. It is sufficient to prove that T>T. We proceed by contradiction and

Acknowledgment

The research of D.Q. Khai was supported by Institute of Mathematics, Vietnam Academy of Science and Technology under grant number IM-VAST01-2020.02. The authors would like to thank the Reviewers and the Editor for the constructive comments and suggestions which helped to improve its content.

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