2001 **Abstract:** We consider the solutions lying on the global attractor of the two-dimensional Navier-Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation, and reaction-diffusion equations with analytic nonlinearities.

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“…However, Foias & Prodi (1967) conjectured that solutions of the Navier-Stokes equations are determined uniquely by a finite number of spatial modes at each time. This conjecture was refined by Foias & Temam (1984), Constantin (1985) and Constantin, Foias & Temam (1988) before being proved in the case of forced two-dimensional turbulence with periodic boundary conditions by Friz & Robinson (2001). These results were recently extended to three-dimensional flow past a body by Galdi (2006), who showed that such flows are uniquely determined by the velocity at a finite number of nodal points near the body.…”

confidence: 90%

“…However, Foias & Prodi (1967) conjectured that solutions of the Navier-Stokes equations are determined uniquely by a finite number of spatial modes at each time. This conjecture was refined by Foias & Temam (1984), Constantin (1985) and Constantin, Foias & Temam (1988) before being proved in the case of forced two-dimensional turbulence with periodic boundary conditions by Friz & Robinson (2001). These results were recently extended to three-dimensional flow past a body by Galdi (2006), who showed that such flows are uniquely determined by the velocity at a finite number of nodal points near the body.…”

confidence: 90%

“…In addition, the best mathematical upper bounds exist only for the two-dimensional Navier-Stokes equations (e.g. Friz & Robinson 2001;Jones & Titi 1993). Note that, due to the lack of vortex stretching, two-dimensional turbulence is usually considered to be less intermittent than threedimensional turbulence.…”

confidence: 99%

“…We point out here that for the 2D Navier-Stokes system with analytic forcing the results of [19], [20] provide the existence of a finite number N of instantaneously determining nodes comparable with the fractal dimension of the attractor. These nodes, however, can be chosen arbitrarily (up to a subset of Ω N with 2N -dimensional Lebesgue measure zero) and therefore do not naturally define a regular lattice of determining nodes.…”

mentioning

confidence: 78%

“…В 1984 г. Ч. Фойяш и Р. Темам [3] высказали предположение, что для си-стемы Навье-Стокса с ограниченной фундаментальной областью Ω ⊂ R 2 эле-менты u(x) аттрактора A однозначно определяются значениями u(x i ) в неко-торой конечной совокупности точек (узлов) x i ∈ Ω. Недавно эта гипотеза была подтверждена [4], [5] в случае, когда вынуждающая сила принадлежит одно-му из функциональных классов Жевре. Интересно, что для периодической области Ω число определяющих узлов x i оказалось соизмеримым с фракталь-ной размерностью аттрактора A. Методы, изложенные в [4], [5], применимы ко многим параболическим уравнениям в частных производных с аналитиче-ским (состоящим из вещественно-аналитических функций) глобальным аттрак-тором.…”

“…Интересно, что для периодической области Ω число определяющих узлов x i оказалось соизмеримым с фракталь-ной размерностью аттрактора A. Методы, изложенные в [4], [5], применимы ко многим параболическим уравнениям в частных производных с аналитиче-ским (состоящим из вещественно-аналитических функций) глобальным аттрак-тором. Данные методы, однако, не приводят к оценкам E-нормы u − v для элементов u, v ∈ A через значения |u(x i ) − v(x i )|.…”