On the regularity of the solutions to the Navier–Stokes equations via the gradient of one velocity component

We improve a regularity criterion for the solutions to the Navier–Stokes equations in the full three-dimensional space involving the gradient of one velocity component. Revising the method used in Pokorný and Zhou (2009, 2010), we show that a weak solution uu is regular on (0,T)(0,T) provided that ∇u3∈Lt(0,T;Ls)∇u3∈Lt(0,T;Ls), where 2/t+3/s=19/102/t+3/s=19/10 for s∈[30/19,10/3]s∈[30/19,10/3] and 2/t+3/s=7/4+1/(2s)2/t+3/s=7/4+1/(2s) for s∈[10/3,∞]s∈[10/3,∞]. It improves the known results for s∈[30/19,150/77)s∈[30/19,150/77) and s∈(10/3,∞]s∈(10/3,∞].