Advances and Challenges in the Solution of Stochastic Partial Differential Equation

The existence of stochastic processes is discussed, describing turbulent solutions of the full Navier-Stokes equation, driven by unidirectional flow, in dimensions one, two and three. These solutions turn out to have a finite velocity and velocity gradient but they are not smooth instead the velocity is Hölder continuous with a Hölder exponent depending on the dimension. They scale with the Kolmogorov scaling in three dimensions and the Batchelor-Kraichnan scaling in two dimension. In one dimensions they scale with the exponent 3/4, that is related to Hack’s law of river basins; stating that the lenght of the main river, in mature river basins, scales with the area of the basin l ∼ Ah. h = 0.58 being Hack’s exponent. The existence of these turbulent solutions is then used to proof the existence of an invariant measure in dimensions one, two and three. The invariant measure characterizes the statistically stationary state of turbulence and it can be used to compute the statistically stationary quantities. These include all the deterministic properties of turbulence and everything that can be computed and measured. In particular, the invariant measure determines the probability density of the turbulent solutions and this can be used to develop accurate sub-grid models in computations of turbulence, bypassing the problem that three-dimensional turbulence cannot be fully resolved with currently existing computer technology.