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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.

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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