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To the best of our knowledge, a direct mapping of a deterministic partial differential equation (PDE) with spatiotemporal chaos onto a stochastic PDE has been carried out only for the Kuramoto-Sivashinsky (KS) equation. This mapping uses a numerical coarse-graining procedure and shows, both in one and two spatial dimensions, that the KS equation is in the universality class of the Kardar-Parisi-Zhang (KPZ) equation, i.e., the long-distance and long-time behaviours of their correlations functions are the same. (See: S Zaleski, Physica D34, 427 (1989); F Hayot, C Jayaprakash and Ch Josserand, Phys. Rev. E47, 911 (1993); C Jayaprakash, F Hayot and R Pandit, Phys. Rev. Lett. 71, 15 (1993).) The case of the deterministically forced Navier-Stokes equation is considerably more subtle; however, in the absence of a direct mapping, it has been conjectured that the appropriate PDA is the NS equation with an additive, Gaussian white noise whose variance has a power-law dependence on the wave-vector [16,17,18]
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