On large-time energy concentration in solutions to the Navier-Stokes equations in the whole 3D space

0380334 - ÚH 2013 RIV DE eng J - Journal Article
Skalák, Zdeněk
On large-time energy concentration in solutions to the Navier-Stokes equations in the whole 3D space.
ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik. Roč. 92, č. 10 (2012), s. 801-815. ISSN 0044-2267. E-ISSN 1521-4001
R&D Projects: GA AV ČR IAA200600801
Institutional research plan: CEZ:AV0Z20600510
Keywords : Navier-Stokes equations * large-time behavior of solutions * energy concentration
Subject RIV: BA - General Mathematics
Impact factor: 0.948, year: 2012

In the first part of the paper we study the large-time behavior of the higher-order space derivatives of solutions to the Navier-Stokes equations in the whole three-dimensional space. We prove that L²- norm of any such derivative decreases quickly than L²- norm of the solution itself. Further, the decay of space derivatives in L²- norm on small time intervals is limited from above by a positive constant. In the second part of the paper we derive several consequences of the results mentioned in the previous paragraph concerning the large-time energy concentration. We show that the energy of any turbulent solution concentrates asymptotically in frequencies from an annulus or a ball centered in the beginning of the coordinates.
Permanent Link: http://hdl.handle.net/11104/0211067