Decaying Turbulence in Generalised Burgers Equation
Résumé
We consider the 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. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. Our proof uses a quantitative version of arguments by Aurell, Frisch, Lutsko and Vergassola \cite{AFLV92}. Our estimates remain valid in the inviscid limit.