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A note on the regularity criterion in terms of pressure for the Navier-Stokes equations. (English) Zbl 1182.35179

Summary: We establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in \(\mathbb R^d\). Here we call \(u\) a Leray weak solution if \(u\) is a weak solution of finite energy, i.e.
\[ u\in L^\infty\big((0,T);L^2\big)\cap L^2\big((0,T);\dot H^1\big). \]
It is known that if a Leray weak solution u belongs to
\[ L^{\frac{2}{1-r}}\big((0,T);L^{\frac dr}\big) \quad\text{for some }0\leq r\leq 1,\tag{1} \]
then \(u\) is regular [see J. Serrin, Arch. Ration. Mech. Anal. 9, 187–195 (1962; Zbl 0106.18302)]. We succeed in proving the regularity of the Leray weak solution \(u\) in terms of pressure under the condition
\[ p\in L^{\frac{2}{1-r}}\big((0,T);\dot X_r(\mathbb R^d)^d\big),\tag{2} \]
where \(\dot X_r(\mathbb R^d)\) is the multiplier space (a definition is given in the text) for \(0\leq r\leq 1\). Since this space \(\dot X_r\) is wider than \(L^{\frac dr}\), the above regularity criterion (2) is an improvement on Y. Zhou’s result [Proc. Amer. Math. Soc. 134, No. 1, 149–156 (2006; Zbl 1075.35044)].

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B45 A priori estimates in context of PDEs
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