Decaying turbulence for the fractional subcritical Burgers equation

TL;DR

This paper extends sharp turbulence estimates for the fractional Burgers equation in the subcritical range, generalizing previous results from the classical case to fractional dissipation, revealing how key quantities scale with viscosity.

Contribution

It provides the first sharp estimates for Sobolev norms and turbulence quantities for the fractional Burgers equation with  in (1,2), generalizing classical results to fractional dissipation.

Findings

Sharp bounds for time-averaged Sobolev norms as functions of viscosity.

Analogues of turbulence quantities scale as powers of separation and wavenumber.

Estimates are similar to the classical case, with  replaced by /(  -1 ).

Abstract

We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +\nu \Lambda^{\alpha} u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is strongly convex and satisfies an additional growth condition, $\Lambda=\sqrt{-\Delta}$, $\nu$ is small and positive, while $\alpha \in (1,\ 2)$ is a constant in the subcritical range. For solutions $u$ of this equation, we generalise the results obtained for the case $\alpha=2$ (i.e. when $-\Lambda^{\alpha}$ is the Laplacian) in [10]. We obtain sharp estimates for the time-averaged Sobolev norms of $u$ as a function of $\nu$. These results yield sharp 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 $\mathbf{k}$ in the Fourier space. The form of all estimates is the same as in the case $\alpha=2$; the only thing that changes (except implicit constants) is that $\nu$ is replaced by $\nu^{1/(\alpha-1)}$.