Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock development? Relatedly, can you see turbulent behavior in solutions to linear approximations of any of the non-linear PDE which are known to behave turbulently in some cases?

Since turbulence is difficult to define rigorously and remains poorly understood mathematically, I will simply use the term to refer to the heuristic phenomena identified as such. I refer to the phenomenon as it is seen in Navier-Stokes and other equations modeling fluid dynamics. I am interested in numerical and theoretical results.