Multidimensional potential Burgers turbulence

We consider the multidimensionall generalised Burgers equation in the space-periodic setting: $$ \partial \mathbf{u}/\partial t+(\nabla f(\mathbf{u}) \cdot \nabla) \mathbf{u} -\nu \Delta \mathbf{u}= \nabla \eta,\quad t \geq 0,\ \mathbf{x} \in T^d=(R/Z)^d, $$ under the assumption that $\mathbf{u}$ is a gradient. 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. \\ \indent For solutions $\mathbf{u}$ of this equation, we study Sobolev norms of $\mathbf{u}$ averaged in time and in ensemble: each of these norms behaves as a given negative power of $\nu$. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which are respectively the separation $\mathbf{l}$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in $\nu$. \\ \indent This paper generalises the results obtained for the one-dimensional case in \cite{BorW} and confirms the physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates does not depend on the dimension: the powers of $\nu, |\mathbf{k}|, |\mathbf{l}|$ are the same in the one- and the multi-dimensional setting.