Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

TL;DR

This paper derives precise bounds for Sobolev norms of solutions to a stochastic Burgers equation, providing insights into turbulence characteristics in a quasi-stationary regime with small viscosity.

Contribution

It presents sharp upper and lower bounds for Sobolev norms of solutions, uniformly in viscosity, in a stochastic Burgers equation, advancing understanding of turbulence in this context.

Findings

Sharp bounds for Sobolev norms in turbulence regime

Results are uniform in viscosity and initial conditions

Provides rigorous estimates aligned with physical turbulence theories

Abstract

We consider the non-homogeneous generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1. Here f is strongly convex and satisfies a growth condition, \nu is small and positive, while \eta is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to t>=T_1, where T_1 depends only on f and on the distribution of \eta. We obtain sharp upper and lower bounds for Sobolev norms of $u$ averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for t>=T_1, and hold uniformly in \nu. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article.