2 00 5 A note on the forced Burgers equation
نویسنده
چکیده
We obtain the exact solution for the Burgers equation with a time dependent forcing, which depends linearly on the spatial coordinate. For the case of a stochastic time dependence an exact expression for the joint probability distribution for the velocity fields at multiple spatial points is obtained. A connection with stretched vortices in hydrodynamic flows is discussed. Burgers equation [1], [2], [3] has extensively been investigated as a model equation for the study of the problem of turbulence. The homogeneous equation can be exactly solved by a Hopf-Cole transformation [4], [5] such that many properties of the spatio-temporal behaviour can be obtained by analytical means. In recent years, the stochastically forced Burgers equation has attracted much interest due to the work of Polyakov [6]. He considered a white noise forcing f (x, t) with spatial correlations < f (x, t)f (x ′ , t) >= κ(x − x ′)δ(t − t ′). This equation has been found to exhibit scaling behaviour. Polyakov suggested to use an operator product expansion in order to assess the scaling behaviour of the velocity increments. The purpose of the present note is to present the general solution of the forced Burgers equation [ ∂ ∂t + v(r, t) ∂ ∂r ]v(r, t) = ν ∂ 2 ∂r 2 v(r, t) + G(t)r , (1) where G(t) is an arbitrary function of time t. As we shall discuss below,
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