Some stability problem for the Navier–Stokes equations in the periodic case

W. M. Zajączkowski¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland and Institute of Mathematics and Cryptology Cybernetics Faculty Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 46 (2019) no. 2, pp. 155-173

Résumé

The Navier–Stokes motions in a box with periodic boundary conditions are considered. First the existence of global regular two-dimensional solutions is proved. Since the external force does not decay in time, the solution has the same property. The necessary estimates and existence are proved step by step in time. Dissipation in the Navier–Stokes equations makes this approach possible. Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of the two-dimensional problems we prove existence of global three-dimensional regular solutions which remain close to the two-dimensional solutions for all time.

Détail

DOI : 10.4064/am2309-8-2018

Keywords: navier stokes motions box periodic boundary conditions considered first existence global regular two dimensional solutions proved since external force does decay time solution has property necessary estimates existence proved step step time dissipation navier stokes equations makes approach possible assuming initial velocity external force sufficiently close initial velocity external force two dimensional problems prove existence global three dimensional regular solutions which remain close two dimensional solutions time