Regularity for the stationary Navier-Stokes equations in bounded domain

Abstract

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain Ω⊑ℝ^N, 5≤N <∞. If u, p satisfy the additional conditions

$$\begin{gathered} ({\text{A}}){\text{ }}\left( {\frac{{u^{\text{2}} }}{{\text{2}}} + p} \right)_ + \in L_{{\text{loc}}}^q (\Omega ),{\text{ }}q \in \left( {\left. {\frac{N}{2},\infty } \right]} \right., \hfill \\ \hfill \\ ({\text{B}}){\text{ }}\int\limits_\Omega {\nabla u\nabla (u\gamma )dx \leqq } {\text{ }}\int\limits_\Omega {\left( {\frac{{u^{\text{2}} }}{{\text{2}}} + p} \right)} {\text{ }}u\nabla \gamma dx + \int\limits_\Omega {fu\gamma dx} {\text{ }}\forall \gamma \in C_0^\infty (\Omega ),{\text{ }}\gamma \geqq 0, \hfill \\ \end{gathered}$$

they become regular. Moreover, it is proved that every weak solution u, p satisfying (A) with q=∞ is regular. The existence of such solutions for N=5 has been established in a former paper [3].