Some new regularity criteria for the Navier–Stokes equations in terms of one directional derivative of the velocity field
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 where is the velocity field, is a scalar pressure and 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 and (see [5], [8]) and then prove that and are uniformly bounded in time. Let is regular on . Since is regular on some positive time interval and is either equal to infinity (in which case the proof is finished) or it is a positive number and is regular on , that is . It is sufficient to prove that . 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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