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This number, denoted by $N$, is defined as the minimum dimension such that for $n\\ge N$, arbitrary $n$-dimensional balls in phase space centred on the solution trajectory $\\omega(x,y,t)$, for $t>0$, contract under the dynamics of the system linearized about $\\omega(x,y,t)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. 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Tran, Luke Blackbourn","submitted_at":"2009-04-20T13:39:21Z","abstract_excerpt":"We derive upper bounds for the number of degrees of freedom of two-dimensional Navier--Stokes turbulence freely decaying from a smooth initial vorticity field $\\omega(x,y,0)=\\omega_0$. This number, denoted by $N$, is defined as the minimum dimension such that for $n\\ge N$, arbitrary $n$-dimensional balls in phase space centred on the solution trajectory $\\omega(x,y,t)$, for $t>0$, contract under the dynamics of the system linearized about $\\omega(x,y,t)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. 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