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Long time decay to the Leray solution of the two‐dimensional Navier–Stokes equations
Author(s) -
Benameur J.,
Selmi R.
Publication year - 2012
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bds033
Subject(s) - mathematics , sobolev space , limit (mathematics) , norm (philosophy) , exponent , mathematical analysis , mathematical proof , infinity , exponential function , differentiable function , initial value problem , exponential growth , homogeneous , pure mathematics , combinatorics , linguistics , philosophy , geometry , political science , law
We give a new proof of the zero limit to the solution of the two‐dimensional Navier–Stokes equations, as time goes to infinity. This proof is done in the frequency space; it is simpler and shorter compared to the existing proofs. Based on this limit, we derive some analytic properties of the solution. Mainly, it becomes infinitely differentiable with respect to time and has value in all Sobolev spaces. Moreover, its regularity grows in an exponential way and its L 2 ( R 2 ) norm decays exponentially fast, as time tends to infinity. Especially, we obtain, for any time t ⩾0, that∫ ξe ( 1 / 2 )ν t| ξ || ℱ ( u ) ( t , ξ ) | 2 d ξ ⩽ ‖ u ( t / 2 ) ‖L 2 ( R 2 ) 2 .We describe the long time behaviour of its homogeneous Sobolev norm for any positive, real exponent, by comparing to usual functions and we ameliorate some existing results. We establish that the Leray solution is stable as time increases.