## Pressure conditions for the local regularity of solutions of the Navier-Stokes equations

### Summary

Summary: We obtain a relationship between the integrability of the pressure gradient and the the integrability of the velocity for local solutions of the Navier--Stokes equations with finite energy. In particular, we show that if the pressure gradient is sufficiently integrable, then the corresponding velocity is locally bounded and smooth in the spatial variables. The result is proven by using De Giorgi type estimates in $L^{\rm weak}_p$ spaces.

35Q30, 76D05

### Keywords/Phrases

Navier-Stokes, regularity, pressure