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An improved regularity criterion for the Navier–Stokes equations in terms of one directional derivative of the velocity field
In this paper, we establish a new multiplicative Sobolev inequality. As applications, we refine and extend the results in Kukavica and Ziane (J Math Phys 48:065203, 2007) and Cao (Discrete Contin Dyn Syst 26:1141–1151, 2010) simultaneously.