Asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations in R3

被引：0

作者：

Skalák Z. ^{[1
]}

机构：

[1] Institute of Hydrodynamics, 166 12 Prague 6

来源：

ANNALI DELL'UNIVERSITA' DI FERRARA | 2009年 / 55卷 / 2期

关键词：

Asymptotic behavior; Energy concentration; Enstrophy concentration; Fast decays; Navier-Stokes equations;

D O I：

10.1007/s11565-009-0073-5

中图分类号：

学科分类号：

摘要：

Let A be the Stokes operator. We show as the main result of the paper that if w is a nonzero global weak solution to the Navier-Stokes equations in R3 satisfying the strong energy inequality, then the energy of the solution w concentrates asymptotically in frequencies smaller than or equal to the finite number (Formula presented.) in the sense that (Formula presented.) for every λ > C(1/2), where {Eλ; λ ≥ 0} is the resolution of identity of A. We also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of w defined as {double pipe}A1/2w{double pipe}. © Università degli Studi di Ferrara 2009.

引用

页码：377 / 394

页数：17

共 10 条

[1]

Kozono H., Ogawa T., Global strong solution and its decay properties for the Navier-Stokes equations in three dimensional domains with non-compact boundaries, Math. Z, 216, pp. 1-30, (1994)

[2]

Okabe T., Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations, J. Differ. Equ, 246, pp. 895-908, (2009)

[3]

Schonbek M.E., Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J, 41, pp. 809-823, (1992)

[4]

Schonbek M.E., Wiegner M., On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinb, 126 A, pp. 677-685, (1996)

[5]

Skalak Z., Some aspects of the asymptotic dynamics of solutions of the homogeneous Navier-Stokes equations in general domains, J. Math. Fluid Mech, (2009)

[6]

Skalak Z., On asymptotic dynamics of solutions of the homogeneous Navier-Stokes equations, Nonlinear Anal, 67, pp. 981-1004, (2007)

[7]

Skalak Z., Conditions for asymptotic energy and enstrophy concentration in solutions to the Navier-Stokes equations, Nonlinear Anal, (2009)

[8]

Skalak Z., Large time behavior of energy in exponentially decreasing solutions of the Navier-Stokes equations, Nonlinear Anal, 71, pp. 593-603, (2009)

[9]

Sohr H., The Navier-Stokes Equations, An Elementary Functional Analytic Approach, (2001)

[10]

Tanabe H., Equations of Evolution, (1979)

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