By Milton Van Dyke
Over four hundred appealing black-and-white images, accrued from around the globe, illustrate the nice range of fluid movement. Flows starting from creeping to hypersonic speeds, in either the laboratory and Nature, are saw without delay, or made obvious utilizing smoke, ink, bubbles, debris, shadographs, schlieren, interferometry, and different innovations. Succinct captions describe the fundamental positive factors of every move.
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Extra info for Album of Fluid Motion
E. if Tr C D 1, Itˆo formula is not valid. However, it is possible to construct solutions as in the two dimensional case. A priori estimates are derived on a translated unknown. However, since uniqueness is not known, we cannot use the same argument as before and cannot construct solutions with respect to the original Wiener process. We only obtain martingale solutions again. Note that, in order to avoid complexity, we did not precise the filtrations above. In fact, all martingales are with respect to the filtration generated by the processes.
Idea of the proof. s. then . \ / is another the dynamics can be restricted to and the measure Q . / invariant measure. t u Proposition 10. Let be an invariant measure then an extremal point in the set of invariant measures. Proof. 7 in . is ergodic if and only if it is t u As already seen, it is in general not very difficult to prove that there exists at least one invariant measure. Thus the set of invariant measures is not empty. It is not difficult to see that it is closed and convex.
D 237(10–12), 1663–1675 (2008) 27. N. Masmoudi, Well-posedness for the FENE dumbbell model of polymeric flows. CPAM 61, 1685–1714 (2008) 28. N. Masmoudi, Global existence of weak solutions to the FENE dumbell model of polymeric flows. Invent. Math. 191(2), 427–500 (2013). Preprint (2010) ¨ 29. C. Ottinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996) 30. F. E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules. Comm. Math. Phys. 277, 729–758 (2008) 31.