## A regularity criterion for the Navier-Stokes equations.(English)Zbl 1220.35111

The authors investigate the regularity of the weak solutions of the incompressible Navier-Stokes equations with viscosity $$\nu$$ $\frac{\partial u}{\partial t} -\nu \triangle u + (u \cdot \nabla) u + \nabla p = f, \qquad \qquad \nabla \cdot u = 0, \tag{1}$ with the initial data $$u_{0} \in L^{2}$$ in $$Q=\Omega \times (0,\infty)$$, and with the periodic or no-slip boundary condition, where $$u_{0}$$ is weakly divergence free and $$\Omega$$ is a open subset of $$\mathbb{R}^{3}$$. The so-called Prodi-Ohyama-Serrin condition plays an important role in the analysis. It consists in the following: Any weak solution $$u$$ of (1) on a cylinder $$B \times (a,b)$$ satisfying $\int_{a}^{b} \left(\int_{B} | u| ^{r} \, dx \right)^{\frac{r'}{r}} dt \,\, < \,\, \infty \quad \text{with} \quad \frac{3}{r} + \frac{2}{r'} < 1, \,\, r \geq 3$ is necessarily a $$L^{\infty }$$ function on any compact subset of the cylinder. It is proven that a weak solution $$u$$ to the Navier-Stokes equations is strong if any two components of $$u$$ satisfy Prodi-Ohyama-Serrin’s criterion. As a local regularity criterion the authors prove that $$u$$ is bounded locally if any two components of the velocity lie in $$L^{6,\infty}$$.

### MSC:

 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids

### Keywords:

Navier-Stokes; Prodi-Ohyama-Serrin condition; regularity
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### References:

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