By Ruth Charney, Michael Davis, Michael Shapiro

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Anal. 196:2 (2002), 323–394. [Treil and Volberg 1997] S. Treil and A. Volberg, “Wavelets and the angle between past and future”, J. Funct. Anal. 143:2 (1997), 269–308. DAMIR Z. il Probability, Geometry and Integrable Systems MSRI Publications Volume 55, 2007 Turbulence of a unidirectional flow ¨ BJORN BIRNIR Dedicated to Henry P. McKean, a mentor and a friend A BSTRACT. We discuss recent advances in the theory of turbulent solutions of the Navier–Stokes equations and the existence of their associated invariant measures.

Spectral functions and the inverse spectral problem are introduced in Section 9. 2. /g; 0 t < d, be a normalized monotonic continuous chain of pairs of entire inner p p mvf’s and let c 2 Ꮿpp . b3t ; b4t I c/. A t /. 0/ D Im . 1) meets these conditions if ˚ pp ¤ ? 4) ˜ P ROOF. 9 in [Arov and Dym 2003a]. 6. 1. 1) in L2 . P ROOF. 8 in [Arov and Dym 2005a]. 2. b4 / as Hilbert spaces with equivalent norms : P ROOF. 8 in [Arov and Dym 2005a]. 3. E/. b4 / are both RKHS’s. 7. 4) is in ˚ pp .

The first range corresponds to a spatial roughness coefficient of one half for small channelizing (very young) landsurfaces. This has been explained, see [Birnir et al. 2007a] and [Edwards and Wilkinson 1982], as Brownian motion of water and sediment over the channelizing surface. The second range with a roughness coefficient of 32 corresponds to the evolution of a young surface forming a convex (geomorphically concave) surface, with young rivers, that evolves by shock formation in the water flow.