Turbulence for the generalised Burgers equation.

*(English. Russian original)*Zbl 1317.35190
Russ. Math. Surv. 69, No. 6, 957-994 (2014); translation from Usp. Mat. Nauk 69, No. 6, 3-44 (2014).

This is a review of results of the author and A. E. Biryuk on turbulence for the generalized Burgers equation \(u_t+f'(u)u_x=\nu u_{xx}+\eta\) considered for space periodic initial data and a random force \(\eta\). Here \(f\) is a smooth and strongly convex function and \(0<\nu\ll 1\) is the viscosity coefficient. The main results include well-posedness of the Cauchy problem and estimates for the Sobolev space norm of \(u\) averaged over time and over the ensemble. They are of the order \(\nu^{-\delta}\) with the same value of \(\delta\geq 0\) for the upper and the lower bounds. These results are interpreted as sharp bounds for small-scale quantities characterizing turbulence. Questions of existence and uniqueness of a stationary measure for randomly forced generalized Burgers equation are also discussed.

Reviewer: Piotr Biler (Wroclaw)

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

35R60 | PDEs with randomness, stochastic partial differential equations |

76F55 | Statistical turbulence modeling |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |