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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.

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)
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