Weinan PDFs for velocity and velocity gradients in Burgers’ turbulence,Comm

We characterize the tails of the probability distribution functions for the solution of Burgers ’ equation with Gaussian initial data and its derivatives ∂ kv(x, t) ∂xk, k = 0, 1, 2.... The tails are “stretched expo-nentials ” of the form P(θ) ∝ exp (−(Re)−p tq θr), where Re is the Reynolds number. The exponents p, q and r depend on the initial spectrum as well as on the order of differentiation, k. These exact results are compared with those obtained using the mapping closure technique. 1