Archive for Rational Mechanics and Analysis

, Volume 128, Issue 4, pp 361–380

# Regularity for the stationary Navier-Stokes equations in bounded domain

• Jens Frehse
• Michael Růžička
Article

## Abstract

Let u, p be a weak solution of the stationary Navier-Stokes equations in a bounded domain Ω⊑ℝN, 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].

## Keywords

Neural Network Complex System Weak Solution Nonlinear Dynamics Bounded Domain
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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