Decaying turbulence for the fractional subcritical Burgers equation

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting:

$\begin{equation} \nonumber\frac{\partial u}{\partial t}+(f(u))_x +ν Λ^{α} u = 0, t ≥ 0,\ \ \ \ {\bf{x}} ∈ {\mathbb{T}}^d = ({\mathbb{R}}/{\mathbb{Z}})^d.\end{equation}$

Here $ f$ is strongly convex and satisfies a growth condition, $ Λ = \sqrt{-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 $ -Λ^{α}$ 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 $ \ell$ in the physical space and the wavenumber $ \bf{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 $ ν^{1/(α-1)}$.