Regularity criteria in terms of the pressure for the Navier-Stokes equations in the critical Morrey-Campanato space (Q628818)

Summary: We establish a Serrin-type regularity criterion in terms of the pressure for Leray weak solutions to the Navier-Stokes equation in \(\mathbb R^3\). It is proved that the solution is regular if the associate pressure satisfies \[ p\in L^{\frac{2}{2-r}} \big((0,T);\dot{\mathcal M}_{2,\frac3r}(\mathbb R^3)\big) \quad\text{or}\quad \nabla p\in L^{\frac{2}{3-r}} \big((0,T);\dot{\mathcal M}_{2,\frac3r}(\mathbb R^3)\big), \] for \(0<r<1\), where \(\dot{\mathcal M}_{2,\frac3r}(\mathbb R^3)\) is the critical Morrey-Campanto space. Regularity criteria for the 3D MHD equations are also given.