Advertisement

Archive for Rational Mechanics and Analysis

, Volume 128, Issue 4, pp 361–380 | Cite as

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Caffarelli, R. Kohn & L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1985), 771–831.Google Scholar
  2. 2.
    J. Frehse, Lecture Notes on Direct Methods and Regularity in the Calculus of Variations, preprint, Universität Bonn (1977).Google Scholar
  3. 3.
    J. Frehse & M. Růžička, On the regularity of the stationary Navier-Stokes equations, Ann. Sci. Norm. Pisa 21 (1994), 63–95.Google Scholar
  4. 4.
    J. Frehse & M. Růžička, Weighted estimates for the stationary Navier-Stokes equations, Acta. Appl. Math, (to appear).Google Scholar
  5. 5.
    J. Frehse & M. Růžička, Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. (to appear).Google Scholar
  6. 6.
    M. Struwe, Regular solutions of the stationary Navier-Stokes on R5, Math. Ann. (to appear).Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Jens Frehse
    • 1
  • Michael Růžička
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonn

Personalised recommendations