1208.5241 (Alexandre Boritchev)

Note on Decaying Turbulence in a Generalised Burgers Equation    [PDF]

Alexandre Boritchev

We consider a generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, where $f$ is strongly convex and $\nu$ is small and positive. Under mild assumptions on the initial condition, we obtain sharp estimates for small-scale quantities. In particular, upper and lower bounds only differ by a multiplicative constant. The quantities which we estimate are the dissipation length scale, small-scale increments and the energy spectrum for solutions $u$, which characterise the \enquote{Burgulence}. We use a quantitative version of arguments contained in a paper by Aurell-Frisch-Lutsko-Vergassola. Note that our estimates still hold in the inviscid limit $\nu \rightarrow 0$.

View original: http://arxiv.org/abs/1208.5241