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Regularity criteria for the Navier-Stokes equations involving the ratio pressure-gradient of velocity. (English) Zbl 1183.35221

Summary: A number of bounds upon the pressure are known to guarantee regularity of the solutions of the Navier-Stokes equations. Since the pressure is the potential whose source is the product of the velocity and its gradient, it is worth to consider bounds depending on the quotient of the pressure and some quantity measuring the size of this source. Estimates involving the ratio pressure-velocity are known. Our result includes the velocity gradient: if the ratio \[ \frac {p}{1+|v|+|\nabla v|^r} \] remains bounded for some \(r<1\), so does the velocity and therefore it retains its regularity.

MSC:

35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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