Y. Le Jan, A.S. Sznitman, Stochastic cascades and 3-dimensional NavierStokes equations, Probab. Theory Rel. Fields 109 (1997), 343-366.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....representations of solutions to Navier Stokes equations, the vortex method, probabilistic model of turbulence, statistical solutions of Foiaslike equations, di usion of passive scalars, stochastic vortex laments. Without the aim to list contributions in all these elds, we mention only [4] 13] [15], 16] 7] Two typical tools, beyond others, are employed: 1) irreducibility, 2) stochastic stationarity. Tool (1) is usually introduced by means of a noise forcing term in the Navier Stokes equations. It is somewhat an idealization of the real behaviour of a uid, but it captures in a sort of ....

Y. Le Jan, A.S. Sznitman, Stochastic cascades and 3-dimensional NavierStokes equations, Probab. Theory Rel. Fields 109 (1997), 343-366.

....have failed. Keywords: Branching processes, Nonlinear di#erential equations, Resummation. 2000 MSC: 60J85) 60J80, 34A34, 34A05 1 Introduction This research started from the desire to understand the limitations on initial conditions imposed by Le Jan and Sznitman in their outstanding work [5] on branching processes representations of the solution to the 3 d Navier Stokes equations. They provide, for small initial conditions, global in time existence and uniqueness of solutions in suitable function spaces. The aim of our paper is to analyse the assumption of small initial conditions ....

Y. Le Jan and A. S. Sznitman. Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Related Fields, 109(3):343--366, 1997. 17

....complicated. In [59] there is given a sketch of an idea for a uniqueness proof in the case where m # 3 and either # = R m , or# is an m dimensional torus, or# is regular, relying on completely di#erent techniques. We also mention that, by means of probabilistic methods, Le Jan and Sznitman [56] established uniqueness and existence results for a class of generalized solutions if# = R 3 . By combining Theorems 6.1, 7.2, and 8.2 we obtain the following existence and uniqueness theorem. For simplicity, we impose more restrictive hypotheses on f than actually needed and leave it to the ....

Y. Le Jan and A.S. Sznitman. Stochastic cascades and 3-dimensional Navier-Stokes equations. Probab. Theory Relat. Fields, 109 (1997), 343--366.

.... global existence (and uniqueness) are known for sufficiently small data; in principle the probabilistic formulation could lead to such results, but we have found some obstacles, so a probabilistic proof of such a result remains an open problem (except for the completely different approach of [25]) The precise local existence and uniqueness result proved here is stated in Section 5 and proved in the subsequent two sections (Sections 6 and 7) The preliminary Section 2 is devoted to a detailed analysis of the equivalence between a system of linear parabolic equations, coupled only in the ....

Y. Le Jan and A.S. Sznitman, Stochastic cascades and 3-dimensional NavierStokes equations, Prob. Th. Rel. Fields 109(3), 343--366, (1997). 62

....for which the bilinear operator does have the expected bicontinuity property. This was observed in different contexts. A somewhat abstract space was constructed in NON STATIONARY NAVIER STOKES WITH AN EXTERNAL FORCE 3 [9,11] a very simple one was introduced by Y. Le Jan and A. S. Sznitman [38,39,14] and, more recently, Y. Meyer announced that the weak L 3 (R 3 ) space, say the Lorentz space L 3;1 (R 3 ) also shares such a property [45] However, it was shown by several authors that even if the bilinear operator B(v; u) t) turns out not to be bicontinuous in a certain limit space ....

Y. Le Jan and A.S. Sznitman, Stochastic cascades and 3-dimensional Navier-Stokes equations, Probab. Theory Related Fields 109 (3), 343-366.

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