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Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations. (English) Zbl 1214.35043

Summary: We establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in d . It is known that if a Leray weak solution u belongs to

L 2 1-r (0,T) ; L d r forsome0r1,

then u is regular. It is proved that if the pressure p associated to a Leray weak solution u belongs to

L 2 1-r (0,T) ; ˙ 2,d r d d ,(*)

where ˙ 2,d r ( d ) d ) is the critical Morrey-Campanato space (a definition is given in the text) for 0<r<1, then the weak solution is actually regular. Since this space ˙ 2,d r is wider than L d r and X ˙ r , the above regularity criterion (*) is an improvement of Zhou’s result.

35Q30Stokes and Navier-Stokes equations
76D05Navier-Stokes equations (fluid dynamics)
76D03Existence, uniqueness, and regularity theory
35B65Smoothness and regularity of solutions of PDE