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

Bae, Hyeong-Ohk; Choe, Hi Jun

Commun. Partial Differ. Equations 32, No. 7, 1173-1187 (2007).

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 \(L^{6,\infty}\).

Reviewer: Jürgen Socolowsky (Brandenburg an der Havel)

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

References:

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