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Maximum Principle and Gradient Estimates for Stationary Solutions of the Navier-Stokes Equations: A Partly Numerical Investigation

In: Advances in Mathematical Fluid Mechanics

Author

Listed:
  • Robert Finn

    (Stanford University, Department of Mathematics)

  • Abderrahim Ouazzi

    (TU Dortmund, Institute for Applied Mathematics)

  • Stefan Turek

    (TU Dortmund, Institute for Applied Mathematics)

Abstract

We calculate numerically the solutions of the stationary Navier-Stokes equations in two dimensions, for a square domain with particular choices of boundary data. The data are chosen to test whether bounded disturbances on the boundary can be expected to spread into the interior of the domain. The results indicate that such behavior indeed can occur, but suggest an estimate of general form for the magnitudes of the solution and of its derivatives, analogous to classical bounds for harmonic functions. The qualitative behavior of the solutions we found displayed some striking and unexpected features. As a corollary of the study, we obtain two new examples of non-uniqueness for stationary solutions at large Reynolds numbers.

Suggested Citation

  • Robert Finn & Abderrahim Ouazzi & Stefan Turek, 2009. "Maximum Principle and Gradient Estimates for Stationary Solutions of the Navier-Stokes Equations: A Partly Numerical Investigation," Springer Books, in: Rolf Rannacher & Adélia Sequeira (ed.), Advances in Mathematical Fluid Mechanics, pages 253-269, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-04068-9_15
    DOI: 10.1007/978-3-642-04068-9_15
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