Abstract
In this paper we describe the results of numerical studies of solutions of the Navier-Stokes System (NSS) under the boundary conditions introduced recently in the paper by Dinaburg et al. (A new boundary problem for the two-dimensional Navier-Stokes system, see this issue of this journal). First, we investigate the decay of Fourier modes, confirming the results and conjectures made in Dinaburg et al. (A new boundary problem for the two-dimensional Navier-Stokes system, see this issue of this journal). Second, we explore the growth of the total energy and enstrophy, which is possible under the adopted boundary conditions. We show that the solutions of the finite-dimensional Galerkin approximations to the NSS may diverge to infinity in finite time, i.e. their energy may blow up.
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References
Dinaburg, E., Li, D., Sinai, Ya.G.: A new boundary problem for the two-dimensional Navier-Stokes system (see this issue of the journal)
Dinaburg, E., Li, D., Sinai, Ya.G.: Navier-Stokes system on the flat cylinder and unit square with slip boundary conditions (submitted)
Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969)
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The author thanks Ya. Sinai and D. Li for involving him in this interesting work, N. Simanyi for help in the analysis of eigensolutions, and A. Korepanov for assistance with Mathematica-based computations. The author was partially supported by NSF grant DMS-0652896.
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Chernov, N. Numerical Studies of a Two-dimensional Navier-Stokes System with New Boundary Conditions. J Stat Phys 135, 751–761 (2009). https://doi.org/10.1007/s10955-009-9759-4
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DOI: https://doi.org/10.1007/s10955-009-9759-4