Decaying turbulence for the fractional subcritical Burgers equation

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: {document}equation ∂ u/∂ t+(f(u))ₓ +ν Λα u = 0, t ≥ 0,\ \ \ \ x ∈ Tᵈ = (R/Z)ᵈ.equation {document} Here f is strongly convex and satisfies a growth condition, Λ = √-Δ, \ ν is small and positive, while $ α ∈ (1,\ 2)$ is a constant in the subcritical range.For solutions u of this equation, we generalise the results obtained for the case $ α = 2$ (i.e. when -Λα is the Laplacian) in [12]. We obtain sharp estimates for the time-averaged Sobolev norms of u as a function of $ ν$. These results yield sharp $ν$-independent estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation in the physical space and the wavenumber k in the Fourier space. The form of all estimates is the same as in the case $ α = 2$; the only thing which changes is that $ ν$ is replaced by ν1/(α-1).