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Invariant measures and a linear model of turbulence

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Invariant measures and a linear model of turbulence RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ANDRZEJ LASOTA Invariantmeasures and a linearmodel of turbulence Rendiconti del Seminario Matematico della Università di Padova, tome 61 (1979), p. 39-48. <http://www.numdam.org/item?id=RSMUP_1979__61__39_0> © Rendiconti del Seminario Matematico della Università di Padova, 1979, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Invariant Measures and a Linear Model of Turbulence. ANDRZEJ LASOTA (*) SUMMARY - A sufficient condition is shown for the existence of continuous measures, invariant and ergodic with respect to semidynamical systems on topological spaces. This condition is applied to a dynamical system generated by a first order linear partial differential equation. 1. Introduction. Roughly speaking a motion of a flow is turbulent if its trajectory in the phase-space is complicated and irregular. There are several ways to make this description precise. The most straightforward one is to give a rigorous definition of turbulent trajectories and then to prove that they exist [1], [3], [7]. Another approach was proposed by G. Prodi in 1960. According to his theory, stationary turbulence occurs when the flow admits a nontrivial ergodic invariant measure [6] (see also [2], [4]). Both points of view are closely related. In fact the existence of turbulent trajectories implies via Kryloff-Bogoluboff theorem the existence of invariant measures, and

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