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Energy bounds for the two-dimensional Navier- Stokes equations in an infinite cylinder (CROSBI ID 200267)

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Slijepčević, Siniša ; Gallay, Thierry Energy bounds for the two-dimensional Navier- Stokes equations in an infinite cylinder // Communications in partial differential equations, 39 (2014), 9; 1741-1769. doi: 10.1080/03605302.2013.870575

Podaci o odgovornosti

Slijepčević, Siniša ; Gallay, Thierry

engleski

Energy bounds for the two-dimensional Navier- Stokes equations in an infinite cylinder

We consider the incompressible Navier-Stokes equation in the cylinder R × T, with no exterior forcing, and we investigate the long- time behavior of solutions arising from merely bounded initial data. Although we do not know if such solutions stay uniformly bounded for all times, we prove that they converge in an appropriate sense to the family of spatially homogeneous equilibria as t goes to infinity. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the L1 norm of the solutions, although our results in this direction are not optimal. Our approach is based on a detailed study of the local energy dissipation in the system, in the spirit of a recent work devoted to a class of dissipative partial differential equations with a formal gradient structure.

Navier-Stokes equation; attractors; limit set; uniform convergence; energy estimates

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Podaci o izdanju

39 (9)

2014.

1741-1769

objavljeno

0360-5302

10.1080/03605302.2013.870575

Povezanost rada

Matematika

Poveznice
Indeksiranost